From: wolf <wolf@0785d39b-7218-0410-832d-ea1e28bc413d>
Date: Wed, 30 Apr 2003 22:27:55 +0000 (+0000)
Subject: Add Annas Nedelec report.
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9de6ec425d94852eb451644c2160d2e2fe0fe7f8;p=dealii-svn.git

Add Annas Nedelec report.


git-svn-id: https://svn.dealii.org/trunk@7519 0785d39b-7218-0410-832d-ea1e28bc413d
---

diff --git a/deal.II/doc/documentation.html b/deal.II/doc/documentation.html
index 2fa0d85a26..f0f5c52c3f 100644
--- a/deal.II/doc/documentation.html
+++ b/deal.II/doc/documentation.html
@@ -193,6 +193,20 @@
               target="body">printable version of the report</a>.
 	      </p>
 
+         <li> <p>
+              A very detailed report on 
+	      <a href="reports/nedelec/main.html"
+                 target="body">Nedelec elements</a> (by Anna
+              Schneebeli, University of Basel, Switzerland). It
+              explains the construction and application of Nedelec
+              edge elements for H-curl spaces, as used, for example,
+              in the numerical solution of the Maxwell equations. It
+              also gives numerical results obtained with deal.II.
+	      Since the report has quite a number of formulas, there
+              is also a <a href="reports/nedelec/nedelec.ps"
+              target="body">printable version of the report</a>.
+	      </p>
+
 	 </ul>
 
 
diff --git a/deal.II/doc/reports/nedelec/images.aux b/deal.II/doc/reports/nedelec/images.aux
new file mode 100644
index 0000000000..c4cb9640ce
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/images.aux
@@ -0,0 +1,2 @@
+\relax 
+\bibstyle{abbrv}
diff --git a/deal.II/doc/reports/nedelec/images.bbl b/deal.II/doc/reports/nedelec/images.bbl
new file mode 100644
index 0000000000..581abba97b
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/images.bbl
@@ -0,0 +1,78 @@
+\begin{thebibliography}{10}
+
+\bibitem{Alonso-Valli}
+A.~Alonso and A.~Valli.
+\newblock An optimal domain decomposition preconditioner for low-frequency
+  time-harmonic {M}axwell equations.
+\newblock {\em Math.~Comp.}, 68(226):607--631, 1999.
+
+\bibitem{Deal}
+W.~Bangerth, R.~Hartmann, and G.~Kanschat.
+\newblock {\em {\tt deal.{I}{I}} Differential Equations Analysis Library,
+  Technical Reference}.
+\newblock IWR, Universit{\"a}t Heidelberg.
+\newblock \texttt{http://www.dealii.org}.
+
+\bibitem{Brezzi-Fortin}
+F.~Brezzi and M.~Fortin.
+\newblock {\em Mixed and Hybrid Finite Element Methods}, volume~15 of {\em
+  Springer Series in Computational Mathematics}.
+\newblock Springer-Verlag, New York, 1991.
+
+\bibitem{Girault-Raviart}
+V.~Girault and P.-A. Raviart.
+\newblock {\em Finite Element Approximation of the Navier-Stokes Equations},
+  volume 749 of {\em Lecture Notes in Mathematics}.
+\newblock Springer-Verlag, Berlin, Heidelberg, 1979, 1981.
+
+\bibitem{Hipt}
+R.~Hiptmair.
+\newblock Finite elements in computational electromagnetism.
+\newblock In {\em Acta Numerica}, pages 1--103. {C}ambridge {U}niversity press,
+  2002.
+
+\bibitem{Monk'92}
+P.~Monk.
+\newblock Analysis of a finite element method for {M}axwell's equations.
+\newblock {\em SIAM J.~Numer.~Anal}, 29:714--729, 1992.
+
+\bibitem{Monk}
+P.~Monk.
+\newblock A simple proof for an edge element discretization of {M}axwell's
+  equations.
+\newblock Submitted for publication. Download version available on Monk's
+  webpage: www.math.udel.edu./~monk, 2001.
+
+\bibitem{Ned1}
+J.~C. N\'ed\'elec.
+\newblock Mixed finite elements in $\mathbb{R}^3$.
+\newblock {\em Numer.~Math.}, 35:315--341, 1980.
+
+\bibitem{Ned3}
+J.~C. N\'ed\'elec.
+\newblock Elements finis mixtes incompressibles pour l'\'equation de {S}tokes
+  dans $\mathbb{R}^3$.
+\newblock {\em Numer.~Math.}, 39:97--112, 1982.
+
+\bibitem{Ned2}
+J.~C. N\'ed\'elec.
+\newblock A new family of mixed finite elements in $\mathbb{R}^3$.
+\newblock {\em Numer.~Math.}, 50:57--81, 1986.
+
+\bibitem{Demko}
+W.~Rachowicz and L.~Demkowicz.
+\newblock A two-dimensional hp-adaptive finite element package for
+  electromagnetics (2{D}hp90\_{E}{M}).
+\newblock Ticam Report 98--16, TICAM, 1998.
+\newblock Download version available on Demkowicz' webpage:
+  www.ticam.utexas.edu/~{L}eszek.
+
+\bibitem{Demko3d}
+W.~Rachowicz and L.~Demkowicz.
+\newblock A three-dimensional hp-adaptive finite element package for
+  electromagnetics (3{D}hp90\_{E}{M}).
+\newblock Ticam Report 00-04.2000, TICAM, 2000.
+\newblock Download version available on Demkowicz' webpage:
+  www.ticam.utexas.edu/~{L}eszek.
+
+\end{thebibliography}
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diff --git a/deal.II/doc/reports/nedelec/images.tex b/deal.II/doc/reports/nedelec/images.tex
new file mode 100644
index 0000000000..1d80be355b
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/images.tex
@@ -0,0 +1,2830 @@
+\batchmode
+
+
+\documentclass[a4paper,11pt]{article}
+\RequirePackage{ifthen}
+
+
+\NeedsTeXFormat{LaTeX2e}
+\usepackage{exscale} 
+\usepackage[dvips]{graphicx}
+\usepackage{shortvrb}  
+\usepackage{amsmath}   
+\usepackage{amssymb}   
+\usepackage{amsfonts}  
+\usepackage{graphicx}	
+
+\addtolength{\topmargin}{-35pt}\addtolength{\headsep}{-2pt}\addtolength{\topskip}{-5pt}\addtolength{\oddsidemargin}{-1.5cm}\addtolength{\evensidemargin}{-1.5cm}\addtolength{\textheight}{45pt}%% less white space at bottom of page
+
+\addtolength{\textwidth}{4cm}%% larger columns
+
+
+%
+\providecommand{\vect}[1]{\underline{#1}}%% vectors%
+\providecommand{\matr}[1]{\mathbf{#1}}%% matrices%
+\providecommand{\ofx}{(\underline{x})}%
+\providecommand{\oftx}{(t,\underline{x})}%
+\providecommand{\R}{\mathbb{R}}%% number sets%
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+\providecommand{\inR}[1]{\in \mathbb{R}^{#1}}%
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+\providecommand{\eye}[1]{\,\mathbb{I}_{#1}\,}%% identity matrix%
+\providecommand{\Laplace}{\Delta}%% Laplace operator%
+\providecommand{\Grad}{\underline{\nabla}}%% Gradient operator%
+\providecommand{\ond}[1]{\in \partial#1}%% on (physical domain .. ) boundary%
+\providecommand{\etime}{\tau^D_{\underline{x}}}%
+\providecommand{\twovec}[2]{\left(\begin{array}{c}#1\\#2\end{array}\right)}%
+\providecommand{\threevec}[3]{\left(\begin{array}{c}#1\\#2\\#3\end{array}\right)}
+
+\parindent 0pt
+
+%
+\providecommand{\Title}[1]{\title{\Large{#1}}  \author{\small{Anna Schneebeli, \today}}\date{}}%
+\providecommand{\Abstract}[1]{\noindent \small \textbf{Abstract:} #1}%
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+\providecommand{\Subsubsection}[1]{\subsubsection{\normalsize{#1}}}%
+\providecommand{\SubsubsectionS}[1]{\subsubsection*{\normalsize{#1}}}
+\newtheorem{remark}{\mdseries{\textsc{Remark}}}\newtheorem{conjecture}{\mdseries{\textsc{Conjecture}}}
+\bibliographystyle{abbrv}
+
+%
+\providecommand{\proof}{\mdseries{\textsc{Proof. }}}%
+\providecommand{\qed}{\begin{flushright} $\square$\  \end{flushright}}
+\newtheorem{definition}{\mdseries{\textsc{Definition}}}\newtheorem{theorem}{\mdseries{\textsc{Theorem}}}\newtheorem{prop}{\mdseries{\textsc{Proposition}}}\newtheorem{example}{\mdseries{\textsc{Example}}}\newtheorem{corollary}{\mdseries{\textsc{Corollary}}}\newtheorem{lemma}{\mdseries{\textsc{Lemma}}}\newtheorem{convention}{\mdseries{\textsc{Convention}}}
+\title{\Large {An $H(\mathop{\rm curl};\Omega )$-conforming FEM: \\
+		N\'ed\'elec's elements of first type}}  \author{\small{Anna Schneebeli, \today}}\date{}
+
+
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+$ \varphi \in \mathcal{D}(\overline{\Omega })$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5477}%
+$\displaystyle \mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_1   \quad \mathrm{and} \quad \mathop{\underline{\rm curl}}\varphi := \left(\begin{array}{c} \partial _y\varphi \\  -\partial _x\varphi \end{array}
+\right) \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5479}%
+$ \mathop{\rm curl}\mathop{\rm curl}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5481}%
+$ \mathop{\underline{\rm curl}}\mathop{\rm curl}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5484}%
+$ \mathop{\rm curl}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5486}%
+$ \underline v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5488}%
+$ \mathop{\underline{\rm curl}}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5490}%
+$ \varphi $%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5492}%
+$\displaystyle \boldsymbol{R} = \left(\begin{array}{cc}
+0 & 1 \\ 
+-1 & 0
+\end{array}\right) \,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5494}%
+$\displaystyle \mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right) 
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5496}%
+$\displaystyle \mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5500}%
+$ \underline t = \boldsymbol{R}^T\underline n$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5504}%
+$ d=3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5506}%
+$ \underline v \in [\mathcal{D}(\overline{\Omega })]^3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5508}%
+$\displaystyle \mathop{\rm curl}\underline v := \nabla \wedge \underline v := \left(\begin{array}{c} 
+\partial _y v_3 - \partial _z v_2 \\ 
+\partial _z v_1 - \partial _x v_3 \\ 						 
+\partial _x v_2 - \partial _y v_1						 
+\end{array} \right)
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5513}%
+$ \tilde{d}=1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5517}%
+$ \tilde{d}=3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5521}%
+$\displaystyle H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d:  \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5525}%
+$\displaystyle (\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\underline v, \underline u)_{L^2(\Omega )} + (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5539}%
+$ [\mathcal{D}(\overline{\Omega })]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5546}%
+$ \underline u$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5548}%
+$ [H(\mathop{\rm curl};\Omega )]^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5552}%
+$ H^1(\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5554}%
+$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx  = \int_{\Omega } \underline u \cdot \mathop{\underline{\rm curl}}\varphi \,dx  + \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5558}%
+$ [H(\mathop{\rm curl};\Omega )]^3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5562}%
+$ [H^1(\Omega )]^3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5564}%
+$\displaystyle \int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \,dx  = \int_{\Omega } \underline u \cdot \mathop{\rm curl}\underline v \,dx  + \int_{\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5566}%
+$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5568}%
+$\displaystyle \mathrm{div}\, (\underline u \wedge \underline v) = \underline v \cdot \mathop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v 
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5570}%
+$ (\underline a\wedge\underline b)\cdot \underline c$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5572}%
+$\displaystyle \int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - \underline u\cdot \mathop{\rm curl}\underline v  \, dx = \int_{\Omega } \mathrm{div}\, (\underline u \wedge \underline v) \, dx
+= \int_{\partial \Omega } (\underline u \wedge \underline v)\cdot \underline n \,ds = \int_{\partial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5574}%
+$ H(\mathop{\rm curl})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5581}%
+$ \underline n$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5591}%
+$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5595}%
+$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5599}%
+$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5603}%
+$ [H^1(\Omega )]^{\tilde{d}}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5605}%
+$ [H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5610}%
+$\displaystyle H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mathop{\rm curl};\Omega ): \quad \underline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\}
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5615}%
+$ [\mathcal{D}(\Omega )]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5617}%
+$ H_0(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5622}%
+$ K_-$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5624}%
+$ K_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5626}%
+$ \mathbb{R}^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5628}%
+$ e = \partial K_-\cap\partial K_+ \neq \emptyset$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5630}%
+$ \Omega = \partial K_-\cup\partial K_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5632}%
+$ v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5636}%
+$ v_-$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5642}%
+$ H(\mathop{\rm curl}; K_-)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5644}%
+$ v_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5650}%
+$ H(\mathop{\rm curl}; K_+)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5652}%
+$ e$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5654}%
+$ v_-\wedge n_- + v_+\wedge n_+ = 0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5658}%
+$ H^{\frac{1}{2}}_{00}(e)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5662}%
+$ H^{\frac{1}{2}}(\partial \Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5674}%
+$ H^{-\frac{1}{2}}(e)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5676}%
+$ \underline v_-$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5678}%
+$ \underline v_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5680}%
+$ L^2(e)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5684}%
+$ H^1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5694}%
+$ \underline u \in H_0(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5696}%
+$ \underline v \in H_0(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5698}%
+$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx =  \int_{\Omega } \underline f\, \cdot \underline v\,dx$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath5701}%
+\begin{displaymath}\begin{split} 	a(\underline u,\underline v) &:= \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx \\  	l(\underline v) & := \int_{\Omega } \underline f\, \cdot \underline v\,dx \end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5703}%
+$ a(\cdot,\cdot)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5705}%
+$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5717}%
+$ \hat{K}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5719}%
+$ F_K(\hat{x})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5721}%
+$ K = F_K(\hat{K})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5723}%
+$ \hat{R}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5727}%
+$ R_K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5729}%
+$ K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5731}%
+$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5735}%
+$ N < \infty$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5739}%
+$ \mathcal{A}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5741}%
+$ \alpha_i(\cdot)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5749}%
+$ \hat{R} = \mathcal{R}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5751}%
+$ \mathbb{P}_k(\hat{\Sigma})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5753}%
+$ k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5755}%
+$ \hat{\Sigma}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5759}%
+$ \tilde{\mathbb{P}}_k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5765}%
+$ d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5770}%
+$\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d : \underline p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5772}%
+$ \hat{x} \in \hat{K}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5778}%
+$ k(k+2)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5785}%
+$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath5787}%
+\begin{displaymath}\begin{split} 					\mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \textrm{for} \quad d=2\,, \\  					\mathrm{dim} (\mathcal{R}^k) &= \frac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. 				\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5789}%
+$ \mathcal{R}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5791}%
+$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\  -\hat{x}_1 				\end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5795}%
+$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\  -\hat{x}_1 \end{array}\right) \subseteq \mathcal{S}^k
+%%\left\{\v p\quad \big|\quad \v p = \tilde{p} \left(\begin{array}{cc} x_2 \\-x_1 \end{array}\right)\,,\, \tilde{p}
+				$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5797}%
+$ \tilde{\mathbb{P}}_{k-1}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5799}%
+$ k-1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5803}%
+$ \mathcal{S}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5809}%
+$ k=1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5811}%
+$ k=2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5814}%
+$\displaystyle \mathcal{R}^1 = \left\langle   								\left(\begin{array}{cc} 1 \\  0 \end{array}\right)\,_, 								\left(\begin{array}{cc} 0 \\  1 \end{array}\right)\,_, 								\left(\begin{array}{cc} \hat{x}_2 \\  -\hat{x}_1 \end{array}\right) \right\rangle$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5816}%
+$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus 
+\left\langle 
+\left(\begin{array}{cc} \hat{x}_1\,\hat{x}_2 \\  -{\hat{x}_1}^2 \end{array}\right)\,_,
+\left(\begin{array}{cc} {\hat{x}_2}^2 \\  -\hat{x}_1\,\hat{x}_2 \end{array}\right)
+\right\rangle					
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5823}%
+$ \mathcal{S}^1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5825}%
+$ \underline p$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5827}%
+$ (\mathbb{P}_{1}(\hat{K}))^3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5829}%
+$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5835}%
+$\displaystyle \underline p \cdot \hat{\underline x} = 
+\sum_{i=1}^3 a_{ii}\hat{x}_i^2 + \sum_{\substack{i,j=1 \\  j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath5839}%
+\begin{displaymath}\begin{split} 					&a_{11}=a_{22}=a_{33} = 0 \\  					&a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,. 				\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5843}%
+$ a_{ij} = 1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5845}%
+$ i=1,2,3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5847}%
+$ j>i$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5849}%
+$\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus  													 \left\langle  													 	\left(\begin{array}{ccc} 0 \\  \hat{x}_3 \\  \hat{x}_2 \end{array}\right)\,_,\, 														\left(\begin{array}{ccc} \hat{x}_3 \\  0 \\  \hat{x}_1 \end{array}\right)\,_,\, 														\left(\begin{array}{ccc} \hat{x}_2 \\  \hat{x}_1 \\  0 \end{array}\right) 														\right\rangle$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5853}%
+$ (\mathbb{P}_{k}(\hat{K}) )^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5868}%
+$ n$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5870}%
+$ n+k+2 \choose n$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5875}%
+$ \hat{\underline t}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5883}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall \hat{\varphi }  						\in \mathbb{P}_{k-1}(\hat{e})\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5885}%
+$ \hat{e}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5889}%
+$ 3k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5891}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi }  						\in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5893}%
+$ k(k-1)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5900}%
+$ \hat{\underline n}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5915}%
+$ 6k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5917}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \quad  						\forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5919}%
+$ \hat{f}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5923}%
+$ 4k(k-1)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5925}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad  						\forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5927}%
+$ \frac{k(k-1)(k-2)}{2}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5933}%
+$ k\leq3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5944}%
+$ \hat{\underline u}\in \mathcal{R}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5946}%
+$ \hat{\alpha}(\hat{\underline u})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5949}%
+$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5951}%
+$ \hat{e}_0 = \overline{(0,0),(1,0)}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5953}%
+$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\  0 \end{array}\right)\,,\quad
+\hat{\underline t}_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} -1 \\  1 \end{array}\right)\,,\quad
+\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\  -1 \end{array}\right)\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5955}%
+$ \mathcal{R}^1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5961}%
+$ \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5963}%
+$ \varphi \equiv 1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5965}%
+$ \mathbb{P}_{0}(\hat{e}_i)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5967}%
+$\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5969}%
+$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5973}%
+$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5975}%
+$ \hat{\underline N}_i$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5979}%
+$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\  \hat{x} \end{array}\right)\,,\quad 					\hat{\underline N}_1 = \left(\begin{array}{cc} -\hat{y} \\  \hat{x} \end{array}\right)\,,\quad 					\hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\  \hat{x}-1 \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5984}%
+$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5988}%
+$ N_i$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline5992}%
+$ \hat{N}_i$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay5996}%
+$\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6000}%
+$ H(\mathop{\rm curl};\hat{K})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6002}%
+$ H(\mathop{\rm curl}; K)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6011}%
+$ \underline N_i(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6015}%
+$\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) =  \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6017}%
+$ \hat{D}F_K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6019}%
+$ \frac{d}{d\hat{x}}F_K(\hat{x})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6022}%
+$ H(\mathop{\rm div}; \Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6026}%
+$ F_K(\hat{x}) = B_K \hat{x} + b_k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6030}%
+$ B_K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6032}%
+$\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) =  B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6035}%
+$ \Omega \subset\mathbb{R}^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6041}%
+$\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6043}%
+$ R$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6048}%
+$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6050}%
+$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6052}%
+$ \hat{x} = 
+F_K^{-1}(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6054}%
+$ F_K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6058}%
+$ D\underline v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6060}%
+$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6062}%
+$\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6074}%
+$ R\,Dv$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6076}%
+$ B_K^{-T}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6080}%
+$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6085}%
+$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6087}%
+$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underline u \, dx = | B_K |^{-1}\,\int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, 						\widehat{\mathop{\rm curl}}\hat{\underline u}  \,d\hat{x}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6092}%
+$ \hat{\underline v}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6096}%
+$ \mathop{\rm Curl}v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6098}%
+$\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6100}%
+$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6102}%
+$\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6109}%
+$ \underline v(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6113}%
+$ \hat{\underline v}(\hat{x})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6115}%
+$\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6117}%
+$ \mathrm{M_i}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6119}%
+$ D(F_K^{-1}) = B_K^{-1}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6121}%
+$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6123}%
+$\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll}  																(\widehat{\mathop{\rm curl}}\,\hat{v} \circ F_K^{-1})_k (x) & \textrm{if} \quad l=i \\  																(B_K^{-1})_{kl} & \textrm{if} \quad l\neq i 												\end{array} \right.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6125}%
+$\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\mathop{\rm Curl}v})_{23} \\  ({\mathop{\rm Curl}v})_{31}  \\  ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6127}%
+$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6129}%
+$ b_{ij} := (B_K^{-1})_{ij}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6131}%
+$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6135}%
+$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\widehat{\mathop{\rm Curl}} \,\hat{v})_{12}
+-(b_{12}b_{33} - b_{32}b_{13})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{31}
++(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6137}%
+$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm curl}}\, v)_1 & b_{12} & b_{13} \\ 
+(\widehat{\mathop{\rm curl}}\, v)_2 & b_{22} & b_{23} \\ 	
+(\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33} 
+\end{array}\right) \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6148}%
+$\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6150}%
+$ (\mathop{\rm curl}\underline v)_1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6152}%
+$\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6154}%
+$ \det \mathrm{M_1}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6158}%
+$\displaystyle \det  \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det\mathcal{B}^{inv}_{11} 
+-(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det\mathcal{B}^{inv}_{21}
++(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6160}%
+$ \mathcal{B}^{inv}_{ij}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6162}%
+$ 2 \times 2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6164}%
+$ B_K^{-1}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6166}%
+$ A \in \mathbb{R}^{3\times 3}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6168}%
+$\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6170}%
+$ \mathcal{A}_{ij}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6174}%
+$ A$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6178}%
+$ A = B_K^{-1}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6180}%
+$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \mathcal{B}^{inv}_{j1} (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_j
+= (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det \mathcal{B}^{inv}_{11} 
+-(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det \mathcal{B}^{inv}_{21}
++(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det  \mathrm{M_1}\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6183}%
+$ C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6185}%
+$ \hat{C} = [0,1]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6187}%
+$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6191}%
+$ \mathcal{Q}_{l,m}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6193}%
+$ \hat{C}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6195}%
+$ l$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6197}%
+$ \hat{x}_1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6199}%
+$ m$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6201}%
+$ \hat{x}_2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6203}%
+$ \mathcal{Q}_{l,m,n}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6217}%
+$ \hat{x}_3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6221}%
+$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} =  			\left(\begin{array}{cc} \hat{u}_1 \\  \hat{u}_2 \end{array}\right): \quad  						 \hat{u}_1 \in \mathcal{Q}_{k-1,k}\,, 								\hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6223}%
+$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{ccc}  					\hat{u}_1 \\  \hat{u}_2 \\ \hat{u}_3 \end{array}\right):\quad  								\hat{u}_1 \in \mathcal{Q}_{k-1,k,k}\,, 								\hat{u}_2 \in \mathcal{Q}_{k,k-1,k}\,,  								\hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6228}%
+$ \hat{C}\subset \mathbb{R}^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6237}%
+$ \mathcal{P}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6241}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad  			\forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6247}%
+$ 4k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6249}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}}  \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x}\,, \quad  			\forall\, \hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\  \hat{\varphi }_2 \end{array}\right) \,, 			\quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6251}%
+$ 2k(k-1)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6273}%
+$ 12k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6275}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \,,\quad 			\forall \,\hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\  \hat{\varphi }_2 \end{array}\right) \,, 			\quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6281}%
+$ 6\cdot 2k(k-1)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6283}%
+$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}}  \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \,,\quad  			\forall\, \hat{\underline \varphi } = \left(\begin{array}{ccc} \hat{\varphi }_1 \\  \hat{\varphi }_2 \\   			\hat{\varphi }_3\end{array}\right) \,,\quad\hat{\varphi }_1\in\mathcal{Q}_{k-1,k-2,k-2}\,,\quad\hat{\varphi _2}\in\mathcal{Q}_{k-2,k-1,k-2}\,, 			\quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6285}%
+$ 3k(k-1)^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6288}%
+$ [0,1]^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6290}%
+$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\  0 \end{array}\right)\,,\quad
+\hat{\underline t}_1 = \left(\begin{array}{cc} 0 \\  1 \end{array}\right)\,,\quad
+\hat{\underline t}_2 = \left(\begin{array}{cc} -1 \\  0 \end{array}\right)\,, \quad
+\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\  -1 \end{array}\right)\,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6292}%
+$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\  0 \end{array}\right)\,,\quad 		\hat{\underline N}_1 = \left(\begin{array}{cc} 0 \\  \hat{x} \end{array}\right)\,,\quad 		\hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\  0 \end{array}\right)\,,\quad 		\hat{\underline N}_3 = \left(\begin{array}{cc} 0 \\  \hat{x}-1 \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6296}%
+$ F_C(\hat{C})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6300}%
+$ \hat{D}F_C(\hat{x})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6302}%
+$ F_C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6307}%
+$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6315}%
+$ \mathop{\rm curl}\underline v$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6317}%
+$ \widehat{\mathop{\rm curl}}\,\hat{\underline v}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6330}%
+$\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad  x = F(\hat{x})\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6334}%
+$ F$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6338}%
+$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6340}%
+$ D(F^{-1})_{ij}(x)=
+\frac{\partial \hat{x}_i}{\partial x_j}(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6342}%
+$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath6344}%
+\begin{displaymath}\begin{split} 			 \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\  			 \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, 			\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath6346}%
+\begin{displaymath}\begin{split} 			 \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) +  			 						  \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\  			 \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) + 			 						  \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. 			\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6348}%
+$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1} - \frac{\partial v_1}{\partial x_2} = 
+\frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) 
+- \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsubsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6353}%
+$ D(F_C^{-1})(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6355}%
+$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6357}%
+$ i,j = 1,2,3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6359}%
+$ \mathop{\rm Curl}v = Dv^T - Dv$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6361}%
+$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,\hat{D}F_C^{-1}) \circ F_C^{-1})(x)
+= (DF_C^{-1})^T(x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6363}%
+$ B_C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6365}%
+$ \hat{D}F_C(\hat(x))$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6370}%
+$ C = F_C(\hat{C})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6376}%
+$\displaystyle \mathop{\rm curl}\underline v =  				\left(\frac{1}{\det \hat{D}F_C}\,\hat{D}F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6383}%
+$ \mathcal{Q}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6389}%
+$ \mathcal{P}_K$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6393}%
+$ K=F(\hat{K})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6399}%
+$ [0,|e|] \ni s \mapsto \underline x(s) \in e$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6401}%
+$ [0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6411}%
+$ \frac{d \underline x}{ds}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6413}%
+$ \frac{d \hat{\underline x}}{d\hat{s}}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6422}%
+$\displaystyle \underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6424}%
+$ |\hat{e}|$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6426}%
+$ |e|$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6432}%
+$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6434}%
+$ \hat{x}_j = \hat{x}_j(\underline x(s))$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6436}%
+$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6438}%
+$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \underline x}{ds} = \left( \hat{\underline v}_j\frac{\partial \hat{x}_j}{\partial x_i}\right) (x)\frac{dx_i}{ds}
+= \hat{\underline v}_j \frac{d \hat{x}_j}{ds} = \hat{\underline v}_j \frac{d \hat{x}_j}{d\hat{s}} \frac{d\hat{s}}{ds} 
+= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds} 
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6440}%
+$ \frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6451}%
+$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6453}%
+$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline t)\varphi \,ds = 
+\int_{\hat{e}} (\hat{\underline v} \cdot \hat{\underline t}) \hat{\varphi } \, d\hat{s}\, = \hat{\alpha}(\hat{\underline u})\,,
+\qquad \forall\, \hat{\varphi } \in
+\mathbb{P}_{k-1}(\hat{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6455}%
+$ K_- = F_-(\hat{K})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6457}%
+$ K_+ = F_+(\hat{K})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6461}%
+$ \underline N$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6465}%
+$ \underline N_-$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6467}%
+$ \underline N_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6475}%
+$ e_+ =F_+(\hat{e}_i)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6477}%
+$ e_- =F_-(\hat{e}_j)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6479}%
+$ \underline t_+ $%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6485}%
+$ \underline t_- = -\underline t_+$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6491}%
+$ \int_{e_+}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6495}%
+$ \int_{e_-}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6499}%
+$ \underline t_-$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6503}%
+$\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6505}%
+$ \alpha^{[K_+]}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6507}%
+$ \alpha^{[K_-]}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6514}%
+$ \hat{e} \ni \hat{x}(s) := \underline a +
+s\, \hat{\underline t}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6516}%
+$ \hat{\underline p} \in \mathcal{S}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6520}%
+$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6522}%
+$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad \textrm{for } i=1,2,3: 
+\quad \hat{p}_i(\hat{x}) = \prod_{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
+\textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6526}%
+$ \hat{x}(s)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6528}%
+$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} + 
+\hat{\varphi }_{k-1}(s)\,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6530}%
+$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6532}%
+$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} = s^k\,\sum_{i=1}^3\hat{t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
+\hat{\varphi }_{k-1}(s)\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6534}%
+$ s^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6536}%
+$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6543}%
+$ \hat{R} = \mathcal{P}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6547}%
+$ (\hat{\underline v}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6560}%
+$\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_+^{-T} \hat{\underline N}_i \,,  			\qquad \underline N_- := -\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6566}%
+$ \underline v := \mathcal{P}_K(\hat{\underline v})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6570}%
+$ \hat{\underline v} \in \hat{R}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6584}%
+$ (\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6590}%
+$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underline N_-\cdot \underline t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6592}%
+$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6598}%
+$\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \int_{\hat{e}_i}  (\hat{\underline N}_i\cdot \hat{\underline t}_i)\hat{\varphi }\,d\hat{s} = 1 			\qquad \textrm{and} \qquad 			\int_{e_-} (\underline N_- \cdot \underline t_-)\varphi \,ds = -\int_{\hat{e}_j}  (\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6602}%
+$ \hat{\alpha}_j(\hat{\underline 
+v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6604}%
+$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6606}%
+$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j \cdot \hat{\underline t})\,|\hat{e}_j| = ({\underline N}_j \cdot {\underline t}_j)\, |e_j|\,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6608}%
+$ \alpha_j(\underline v)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6610}%
+$ e_j$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6612}%
+$ |e_j|\left(\underline v\cdot \underline t_j\right)|_e$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6615}%
+$ \alpha^{[K]}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6619}%
+$ \underline t = \frac{|\hat{e}|}{|e|}\,(\hat{D}F)\,\hat{\underline t}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6623}%
+$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{subsection}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6626}%
+$ V_h \subset H(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6628}%
+$\displaystyle \| \underline u - \Pi_h^k \underline u\|_{H(\mathop{\rm curl}; \Omega )} = C\,\inf_{w\in V_h}\| \underline u -  \underline w\|_{H(\mathop{\rm curl}; \Omega )}\,,
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6630}%
+$ \Pi_h^k \underline u \in \mathcal{R}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6632}%
+$ \Pi_h^k \underline u \in \mathcal{P}^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6636}%
+$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6638}%
+$ \alpha$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6640}%
+$ \Pi_h^k$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6642}%
+$ \underline v\in H^r(\mathop{\rm curl})$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6644}%
+$ r>\frac{1}{2}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6647}%
+$ \mathcal{T}_h$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6649}%
+$ h>0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6655}%
+$ C>0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6657}%
+$ r$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6659}%
+$ h$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6663}%
+$\displaystyle \| \underline v - \Pi_h^k \underline v\|_{H(\mathop{\rm curl}; \Omega )} \leq C\,h^{\min\{r,k\}} \|\underline v\|_{H^r(\mathop{\rm curl};\Omega )}\,,$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6665}%
+$ \underline v\in H^r(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6675}%
+$ \mathcal{O}(h^k)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6677}%
+$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6679}%
+$\displaystyle \| \underline v - \Pi_h^k \underline v\|_{L^2(\Omega )} \leq C h^k |\underline v|_{H^k(\Omega )}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6681}%
+$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6685}%
+$ H^k(K)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6687}%
+$ \mathcal{R}^k(K)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6691}%
+$ \| \underline u - \underline u_h\|_{L^2(\Omega )}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6696}%
+$ s>\frac{1}{2}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6698}%
+$\displaystyle \| \underline u -  \underline u_h\|_{L^2(\Omega )} \leq C h^s \| \underline u - \underline u_h\|_{H(\mathop{\rm curl}; \Omega )}\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6702}%
+$ s=1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6704}%
+$ [\mathbb{P}_k]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6711}%
+$ \Omega = [-1,1]^d$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6715}%
+$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\  3 - x^2 \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6717}%
+$\displaystyle c\equiv 1 \,, \qquad  		\underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\  -\sin\pi x\cos\pi y \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6721}%
+$ 2^5$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6723}%
+$ 2^{13}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6731}%
+$ \mathcal{O}(h)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6733}%
+$ L^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6735}%
+$ \#$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6818}%
+$ H(\mathop{\rm curl};\Omega )$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap4265}%
+% latex2html id marker 4265
+\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline6830}%
+$ H(\mathop{\rm curl};(\Omega ))$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap4271}%
+% latex2html id marker 4271
+\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap4293}%
+\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay6913}%
+$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{ccc}   xy(1 - y^2)(1-z^2) + 2xy(1-z^2) \\  																	  y^2(1 - x^2)(1-z^2) + (1-y^2)(2-x^2-z^2) \\  																	  yz(1 - x^2)(1-y^2) + 2yz(1-x^2) 											 \end{array}\right)\,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap4329}%
+\includegraphics[width=9.5cm, height=7cm]{field1.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap4335}%
+\includegraphics[width=9.5cm, height=7cm]{field2.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7106}%
+$ \mathbb{R}^3$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\appendix
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7119}%
+$ \mathbb{R}^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7121}%
+$ \varphi (x,y)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7125}%
+$ c>0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7127}%
+$ w$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7129}%
+\begin{displaymath}\begin{split} 			-\Delta w + c\, w &= \varphi \quad \mathrm{in} \quad \Omega \\  			\underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,. 		\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7131}%
+$ \underline E := \nabla^{\perp} w$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7133}%
+\begin{displaymath}\begin{split} 			\underline \mathop{\rm curl}\mathop{\rm curl}\underline E + c\, \underline E  = \underline f \quad \mathrm{in} \quad \Omega \,, \\  			\underline E \wedge \underline n = 0 \quad \mathrm{on} \quad \partial \Omega \,, 		\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7135}%
+$ \underline f := \nabla^{\perp} \varphi $%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7137}%
+$ \nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\  -\partial _x\varphi 
+\end{array}\right)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7141}%
+$\displaystyle \underline E \wedge \underline n = \underline E \cdot \underline t = {\nabla w}^T \boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7143}%
+$ \underline E$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7147}%
+$ \nabla\cdot\nabla^{\perp}w = 0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7151}%
+$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7153}%
+$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7155}%
+$ \nabla^{\perp}w$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7157}%
+$ \nabla^{\perp} \varphi $%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7162}%
+\begin{displaymath}\begin{split} 			-\Delta w  &= \lambda \, w \quad \mathrm{in} \quad \Omega \\  			\underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,, 		\end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7164}%
+$ \varphi = (\lambda + c)\,w$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7166}%
+$ \Omega = [-1,1]^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7168}%
+$ \lambda = 2\pi^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7170}%
+$ w = \cos\pi x\cos\pi y$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7172}%
+$\displaystyle \underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\  -\sin\pi x\cos\pi y \end{array}\right)\,, \qquad
+\underline E = \pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\  -\sin\pi x\cos\pi y \end{array}\right)\,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7179}%
+$ w(x,y) = (1-x^2)^2(1-y^2)^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7181}%
+$ \underline n \cdot \nabla w = 0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7183}%
+$ \partial [-1,1]^2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7185}%
+$ \varphi = -\Delta w + c w$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\appendix
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7188}%
+\begin{displaymath}\begin{split} 	\varepsilon \frac{\partial \mathcal{E}}{\partial t} & = \mathop{\rm curl}\mathcal{H} - \sigma \mathcal{E} \,, \\  	\mu \frac{\partial \mathcal{H}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,, \end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7190}%
+$ \mathcal{E}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7192}%
+$ \mathcal{H}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7194}%
+$ \varepsilon (x), \mu(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7196}%
+$ \sigma(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7202}%
+$ L^{\infty}(\Omega )^{d\times d}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7204}%
+$ \varepsilon (x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7206}%
+$ \mu(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7211}%
+$ \mathcal{E}(x,t)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7213}%
+$ \mathcal{H}(x,t)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7215}%
+\begin{displaymath}\begin{split} 	\mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(i\omega t)\right) \,, \\  	\mathcal{H}(x,t) &= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,. \end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7217}%
+$ E(x), H(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7219}%
+$ \omega\neq 0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7222}%
+$ E(x) \exp(i\omega t)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7224}%
+$ H(x) \exp(i\omega t)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7226}%
+$ H(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7228}%
+$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7230}%
+$ |\omega|$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7232}%
+$\displaystyle \omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,.
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7234}%
+$ \omega^2\varepsilon E(x)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7236}%
+$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0
+$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_indisplay7242}%
+$\displaystyle E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.$%
+\lthtmlindisplaymathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7244}%
+$ \tilde{E}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline7246}%
+$ \underline u = E - \tilde{E}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7248}%
+\begin{displaymath}\begin{split} 	\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) + i\omega\sigma u &= F \quad \mathrm{in} \quad \Omega \,, \\  	u \wedge n &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+
+\end{document}
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+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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+<HEAD>
+<TITLE>An -conforming FEM: N&#233;d&#233;lec's elements of first type</TITLE>
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+
+</HEAD>
+
+<BODY >
+<H1 ALIGN=CENTER><FONT SIZE="+2">An <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM: 
+<BR>
+N&#233;d&#233;lec's elements of first type</FONT></H1>
+<P ALIGN=CENTER><STRONG><FONT SIZE="-1">Anna Schneebeli, April 30, 2003</FONT></STRONG></P>
+<P ALIGN=LEFT></P>
+<FONT SIZE="-1"><B>Abstract:</B> The aim of this report is to give an introduction to N&#233;d&#233;lec's <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method of first 
+type. As the name suggests, this method has been introduced in 1980 by J.&nbsp;C.&nbsp;N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned1">8</A>].
+<BR>
+In the first section, we present the model problem and introduce the framework for its variational formulation.
+<BR>
+In the second section, we present N&#233;d&#233;lec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">. 
+We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector
+fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and
+trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks
+on approximation results.
+<BR>
+Numerical results, which serve to illustrate the convergence of the method, are presented in the third section.
+In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace
+equation.
+<BR>
+In Appendix B we motivate the model problem studied in the report by considering the 
+time-harmonic Maxwell's equations in the low-frequency case. 
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<BR><HR>
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"></A>
+
+<UL>
+<LI><A NAME="tex2html11"
+  HREF="node1.html">1 <FONT SIZE="+1">The model problem and the space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+<UL>
+<LI><A NAME="tex2html12"
+  HREF="node1.html#SECTION00011000000000000000">1.1 Definitions</A>
+<LI><A NAME="tex2html13"
+  HREF="node1.html#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
+<LI><A NAME="tex2html14"
+  HREF="node1.html#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html15"
+  HREF="node2.html">2 <FONT SIZE="+1">N&#233;d&#233;lec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+<UL>
+<LI><A NAME="tex2html16"
+  HREF="node2.html#SECTION00021000000000000000">2.1 Construction of N&#233;d&#233;lec elements on tetrahedral grids</A>
+<UL>
+<LI><A NAME="tex2html17"
+  HREF="node2.html#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html18"
+  HREF="node2.html#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html19"
+  HREF="node2.html#SECTION00021300000000000000">2.1.3 Piola transformation</A>
+<LI><A NAME="tex2html20"
+  HREF="node2.html#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
+<LI><A NAME="tex2html21"
+  HREF="node2.html#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
+</UL>
+<LI><A NAME="tex2html22"
+  HREF="node2.html#SECTION00022000000000000000">2.2 N&#233;d&#233;lec Elements on affine quadrilateral or hexahedral grids</A>
+<UL>
+<LI><A NAME="tex2html23"
+  HREF="node2.html#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html24"
+  HREF="node2.html#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html25"
+  HREF="node2.html#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
+</UL>
+<LI><A NAME="tex2html26"
+  HREF="node2.html#SECTION00023000000000000000">2.3 Construction of N&#233;d&#233;lec elements on bi- or trilinear elements</A>
+<UL>
+<LI><A NAME="tex2html27"
+  HREF="node2.html#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
+<LI><A NAME="tex2html28"
+  HREF="node2.html#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
+</UL>
+<LI><A NAME="tex2html29"
+  HREF="node2.html#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
+<LI><A NAME="tex2html30"
+  HREF="node2.html#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html31"
+  HREF="node3.html">3 <FONT SIZE="+1">Numerical results</FONT></A>
+<LI><A NAME="tex2html32"
+  HREF="node4.html">Bibliography</A>
+<LI><A NAME="tex2html33"
+  HREF="node5.html">A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
+<LI><A NAME="tex2html34"
+  HREF="node6.html">A. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT></A>
+<UL>
+<LI><A NAME="tex2html35"
+  HREF="node6.html#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
+</UL></UL>
+<!--End of Table of Child-Links-->
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
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+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>An -conforming FEM: N&#233;d&#233;lec's elements of first type</TITLE>
+<META NAME="description" CONTENT="An -conforming FEM: N&#233;d&#233;lec's elements of first type">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+<H1 ALIGN=CENTER><FONT SIZE="+2">An <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM: 
+<BR>
+N&#233;d&#233;lec's elements of first type</FONT></H1>
+<P ALIGN=CENTER><STRONG><FONT SIZE="-1">Anna Schneebeli, April 30, 2003</FONT></STRONG></P>
+<P ALIGN=LEFT></P>
+<FONT SIZE="-1"><B>Abstract:</B> The aim of this report is to give an introduction to N&#233;d&#233;lec's <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method of first 
+type. As the name suggests, this method has been introduced in 1980 by J.&nbsp;C.&nbsp;N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned1">8</A>].
+<BR>
+In the first section, we present the model problem and introduce the framework for its variational formulation.
+<BR>
+In the second section, we present N&#233;d&#233;lec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">. 
+We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector
+fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and
+trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks
+on approximation results.
+<BR>
+Numerical results, which serve to illustrate the convergence of the method, are presented in the third section.
+In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace
+equation.
+<BR>
+In Appendix B we motivate the model problem studied in the report by considering the 
+time-harmonic Maxwell's equations in the low-frequency case. 
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<BR><HR>
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"></A>
+
+<UL>
+<LI><A NAME="tex2html11"
+  HREF="node1.html">1 <FONT SIZE="+1">The model problem and the space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+<UL>
+<LI><A NAME="tex2html12"
+  HREF="node1.html#SECTION00011000000000000000">1.1 Definitions</A>
+<LI><A NAME="tex2html13"
+  HREF="node1.html#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
+<LI><A NAME="tex2html14"
+  HREF="node1.html#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html15"
+  HREF="node2.html">2 <FONT SIZE="+1">N&#233;d&#233;lec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+<UL>
+<LI><A NAME="tex2html16"
+  HREF="node2.html#SECTION00021000000000000000">2.1 Construction of N&#233;d&#233;lec elements on tetrahedral grids</A>
+<UL>
+<LI><A NAME="tex2html17"
+  HREF="node2.html#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html18"
+  HREF="node2.html#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html19"
+  HREF="node2.html#SECTION00021300000000000000">2.1.3 Piola transformation</A>
+<LI><A NAME="tex2html20"
+  HREF="node2.html#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
+<LI><A NAME="tex2html21"
+  HREF="node2.html#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
+</UL>
+<LI><A NAME="tex2html22"
+  HREF="node2.html#SECTION00022000000000000000">2.2 N&#233;d&#233;lec Elements on affine quadrilateral or hexahedral grids</A>
+<UL>
+<LI><A NAME="tex2html23"
+  HREF="node2.html#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html24"
+  HREF="node2.html#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html25"
+  HREF="node2.html#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
+</UL>
+<LI><A NAME="tex2html26"
+  HREF="node2.html#SECTION00023000000000000000">2.3 Construction of N&#233;d&#233;lec elements on bi- or trilinear elements</A>
+<UL>
+<LI><A NAME="tex2html27"
+  HREF="node2.html#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
+<LI><A NAME="tex2html28"
+  HREF="node2.html#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
+</UL>
+<LI><A NAME="tex2html29"
+  HREF="node2.html#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
+<LI><A NAME="tex2html30"
+  HREF="node2.html#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html31"
+  HREF="node3.html">3 <FONT SIZE="+1">Numerical results</FONT></A>
+<LI><A NAME="tex2html32"
+  HREF="node4.html">Bibliography</A>
+<LI><A NAME="tex2html33"
+  HREF="node5.html">A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
+<LI><A NAME="tex2html34"
+  HREF="node6.html">A. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT></A>
+<UL>
+<LI><A NAME="tex2html35"
+  HREF="node6.html#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
+</UL></UL>
+<!--End of Table of Child-Links-->
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
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+%%Trailer
+end
+userdict /end-hook known{end-hook}if
+%%EOF
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+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>1 The model problem and the space </TITLE>
+<META NAME="description" CONTENT="1 The model problem and the space ">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
+
+<UL>
+<LI><A NAME="tex2html46"
+  HREF="#SECTION00011000000000000000">1.1 Definitions</A>
+<LI><A NAME="tex2html47"
+  HREF="#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
+<LI><A NAME="tex2html48"
+  HREF="#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
+</UL>
+<!--End of Table of Child-Links-->
+<HR>
+
+<H1><A NAME="SECTION00010000000000000000">
+1 <FONT SIZE="+1">The model problem and the space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+</H1>
+<P>
+<FONT SIZE="-1">Consider the vector-valued model problem in a Lipschitz domain <!-- MATH
+ $\Omega \in \mathbb{R}^d$
+ -->
+<IMG
+ WIDTH="61" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img4.gif"
+ ALT="$ \Omega \in \mathbb{R}^d$">, <IMG
+ WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ d=2,3$">:
+</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:model_problem"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u  = \underline f \quad \mathrm{in} \quad \Omega \,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="246" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img6.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u = \underline f \quad \mathrm{in} \quad \Omega \,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(1)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+with right hand side <!-- MATH
+ $\underline f \in L^2(\Omega )^d$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img7.gif"
+ ALT="$ \underline f \in L^2(\Omega )^d$">. 
+<BR>
+We assume a homogeneous Dirichlet boundary condition on the tangential trace
+</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_PCB"></A><!-- MATH
+ \begin{equation}
+\underline u \wedge \underline n  = 0
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img8.gif"
+ ALT="$\displaystyle \underline u \wedge \underline n = 0$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(2)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+on the boundary <!-- MATH
+ $\partial \Omega$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img9.gif"
+ ALT="$ \partial \Omega $"> of <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $">.
+<BR>
+The coefficient <IMG
+ WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img11.gif"
+ ALT="$ c(x)$"> is assumed to be bounded and uniform positive definite.
+<BR>
+This type of problem typically arises in particular settings of
+Maxwell`s equations. The boundary condition (<A HREF="node1.html#eq:_PCB">2</A>) then applies to a perfectly conducting boundary. 
+For a derivation of the model problem (<A HREF="node1.html#eq:model_problem">1</A>), refer to Appendix <A HREF="node6.html#appendix:_Maxwell_s_Eq.">A</A>.
+<BR></FONT>
+<P>
+<FONT SIZE="-1">The subject of this section is to give an appropriate setting for a variational formulation of (<A HREF="node1.html#eq:model_problem">1</A>).
+<BR>
+A more detailed treatment of the following notions and proofs can be found in [<A
+ HREF="node4.html#Girault-Raviart">4</A>].
+</FONT>
+<P>
+
+<H2><A NAME="SECTION00011000000000000000">
+1.1 Definitions</A>
+</H2>
+<P>
+<P>
+<DIV><A NAME="def:_tangent"><B>C<SMALL>ONVENTION</SMALL>  1</B></A> &nbsp; 
+In the following, the vector <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$"> will denote the unit tangent vector w.&nbsp;r.&nbsp;t. an edge of a triangle or quadrilateral, 
+	oriented counterclockwise with respect to the corresponding triangle or quadrilateral. 
+	(In 3d, the considered triangles or quadrilaterals will always be faces of a polyhedron, and the counterclockwise orientation has to be
+	understood as induced by "outward unit normal of the face, plus right hand rule" ).</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Let us first consider the case of <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$">. For <!-- MATH
+ $\underline v = \left(\begin{array}{c} v_1(x,y) \\v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$
+ -->
+<IMG
+ WIDTH="215" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img14.gif"
+ ALT="$ \underline v = \left(\begin{array}{c} v_1(x,y) \\  v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$">
+and <!-- MATH
+ $\varphi \in \mathcal{D}(\overline{\Omega })$
+ -->
+<IMG
+ WIDTH="79" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img15.gif"
+ ALT="$ \varphi \in \mathcal{D}(\overline{\Omega })$"> we define the scalar- and the
+vector-valued curl-operators:
+</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_1   \quad \mathrm{and} \quad \mathop{\underline{\rm curl}}\varphi := \left(\begin{array}{c} \partial _y\varphi \\-\partial _x\varphi \end{array}
+	\right) \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="396" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img16.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_...
+...in{array}{c} \partial _y\varphi \\  -\partial _x\varphi \end{array}\right) \,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+We note that the <!-- MATH
+ $\mathop{\rm curl}\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img17.gif"
+ ALT="$ \mathop{\rm curl}\mathop{\rm curl}$">-operator in two dimensions has to be understood as <!-- MATH
+ $\mathop{\underline{\rm curl}}\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img18.gif"
+ ALT="$ \mathop{\underline{\rm curl}}\mathop{\rm curl}$">.
+<BR></FONT><P>
+<DIV><A NAME="re:_rotation"><B>R<SMALL>EMARK</SMALL>  1</B></A> &nbsp; 
+In the two dimensional case, the <!-- MATH
+ $\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img19.gif"
+ ALT="$ \mathop{\rm curl}$"> operator is simply the divergence of the rotated field <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">. Similarly, the <!-- MATH
+ $\mathop{\underline{\rm curl}}$
+ -->
+<IMG
+ WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img21.gif"
+ ALT="$ \mathop{\underline{\rm curl}}$"> operator is
+the rotated gradient field of <IMG
+ WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img22.gif"
+ ALT="$ \varphi $">. Setting
+<!-- MATH
+ \begin{displaymath}
+\boldsymbol{R} = \left(\begin{array}{cc}
+				0 & 1 \\
+				-1 & 0
+				\end{array}\right) \,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="144" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img23.gif"
+ ALT="$\displaystyle \boldsymbol{R} = \left(\begin{array}{cc}
+0 &amp; 1 \\
+-1 &amp; 0
+\end{array}\right) \,,
+$">
+</DIV><P></P>
+we have
+<!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img24.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right)
+$">
+</DIV><P></P>
+and
+<!-- MATH
+ \begin{displaymath}
+\mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="122" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img25.gif"
+ ALT="$\displaystyle \mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,.
+$">
+</DIV><P></P>
+We further note that the tangential vector <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$"> is just the rotated outward unit normal vector
+<!-- MATH
+ $\underline t = \boldsymbol{R}^T\underline n$
+ -->
+<IMG
+ WIDTH="72" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img26.gif"
+ ALT="$ \underline t = \boldsymbol{R}^T\underline n$">. This will enable us to derive statements for the <!-- MATH
+ $\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img19.gif"
+ ALT="$ \mathop{\rm curl}$">-operators in two dimensions from statements for the 
+divergence and gradient operators in two dimensions.</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">For the case of <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$"> and a vector field <!-- MATH
+ $\underline v \in [\mathcal{D}(\overline{\Omega })]^3$
+ -->
+<IMG
+ WIDTH="94" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img28.gif"
+ ALT="$ \underline v \in [\mathcal{D}(\overline{\Omega })]^3$"> we write
+</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}\underline v := \nabla \wedge \underline v := \left(\begin{array}{c}
+											\partial _y v_3 - \partial _z v_2 \\
+											\partial _z v_1 - \partial _x v_3 \\
+											\partial _x v_2 - \partial _y v_1						 
+				  						\end{array} \right)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="281" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img29.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v := \nabla \wedge \underline v := \l...
+... - \partial _x v_3 \\
+\partial _x v_2 - \partial _y v_1
+\end{array} \right)
+$">
+</DIV><P></P><P>
+<DIV><B>D<SMALL>EFINITION</SMALL>  1</B> &nbsp; 
+For <IMG
+ WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ d=2,3$"> we write <!-- MATH
+ $\tilde{d}=1$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img30.gif"
+ ALT="$ \tilde{d}=1$"> if <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> and <!-- MATH
+ $\tilde{d}=3$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img31.gif"
+ ALT="$ \tilde{d}=3$"> if <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">, and we define
+<!-- MATH
+ \begin{displaymath}
+H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d:  \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="364" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
+ SRC="img32.gif"
+ ALT="$\displaystyle H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d: \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
+$">
+</DIV><P></P></DIV><P></P>
+<FONT SIZE="-1">
+<!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> endowed with the inner product
+</FONT><!-- MATH
+ \begin{displaymath}
+(\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\underline v, \underline u)_{L^2(\Omega )} + (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="374" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img33.gif"
+ ALT="$\displaystyle (\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\...
+... (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
+$">
+</DIV><P></P><FONT SIZE="-1">
+is a Hilbert space.
+<BR></FONT>
+<P>
+
+<H2><A NAME="SECTION00012000000000000000">
+1.2 Trace theorem, integration by parts</A>
+</H2><FONT SIZE="-1">
+The space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> will be the appropriate Sobolev space for a weak formulation of the model problem. 
+In this section we provide a notion of trace of a <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-function onto the boundary <!-- MATH
+ $\partial \Omega$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img9.gif"
+ ALT="$ \partial \Omega $"> and we define intergation by parts on the
+space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="th:density"><B>T<SMALL>HEOREM</SMALL>  1</B></A> (Approximation Property)  &nbsp; 
+For <IMG
+ WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ d=2,3$">, <!-- MATH
+ $[\mathcal{D}(\overline{\Omega })]^d$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img34.gif"
+ ALT="$ [\mathcal{D}(\overline{\Omega })]^d$"> is dense in <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">.</DIV><P></P>
+<FONT SIZE="-1">
+See [<A
+ HREF="node4.html#Girault-Raviart">4</A>] p.13, p.20 for the proof in the 2d-case and p.20 for a reference to the proof in 3d proposed in Duvaut &amp; Lions, 1971.
+<BR></FONT>
+<P>
+<FONT SIZE="-1">Equipped with this approximation property of smooth functions to elements of <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">, we can state
+</FONT>
+<P>
+<P>
+<DIV><A NAME="eq:_PI"><B>T<SMALL>HEOREM</SMALL>  2</B></A> (Green's Formula)  &nbsp; 
+For the 2d case, let <!-- MATH
+ $\underline u$
+ -->
+<IMG
+ WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img35.gif"
+ ALT="$ \underline u$"> be in <!-- MATH
+ $[H(\mathop{\rm curl};\Omega )]^2$
+ -->
+<IMG
+ WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img36.gif"
+ ALT="$ [H(\mathop{\rm curl};\Omega )]^2$"> and <IMG
+ WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img22.gif"
+ ALT="$ \varphi $"> be a test function in <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$">. We have 
+<!-- MATH
+ \begin{displaymath}
+\int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx  = \int_{\Omega } \underline u \cdot \mathop{\underline{\rm curl}}\varphi \,dx  + \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="380" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img38.gif"
+ ALT="$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx = \...
+...+ \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:,
+$">
+</DIV><P></P>
+
+<P>
+For the 3d case, let <!-- MATH
+ $\underline u$
+ -->
+<IMG
+ WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img35.gif"
+ ALT="$ \underline u$"> be in <!-- MATH
+ $[H(\mathop{\rm curl};\Omega )]^3$
+ -->
+<IMG
+ WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img39.gif"
+ ALT="$ [H(\mathop{\rm curl};\Omega )]^3$"> and <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$"> be a test function in <!-- MATH
+ $[H^1(\Omega )]^3$
+ -->
+<IMG
+ WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img40.gif"
+ ALT="$ [H^1(\Omega )]^3$">. We then have 
+<!-- MATH
+ \begin{displaymath}
+\int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \,dx  = \int_{\Omega } \underline u \cdot \mathop{\rm curl}\underline v \,dx  + \int_{\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="400" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img41.gif"
+ ALT="$\displaystyle \int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \...
+...\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:,
+$">
+</DIV><P></P>
+The boundary integrals are understood as duality pairings in <!-- MATH
+ $[H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$
+ -->
+<IMG
+ WIDTH="181" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img42.gif"
+ ALT="$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$"> .
+<BR></DIV><P></P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>For smooth functions, it is easy to see that the above Green's formula holds. In the 2d case this follows just
+from Gauss' divergence theorem and remark <A HREF="node1.html#re:_rotation">1</A>.
+<BR>
+For the 3d case we use the identity
+</FONT><!-- MATH
+ \begin{displaymath}
+\mathrm{div}\, (\underline u \wedge \underline v) = \underline v \cdot \mathop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="257" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img43.gif"
+ ALT="$\displaystyle \mathrm{div}\, (\underline u \wedge \underline v) = \underline v ...
+...thop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v
+$">
+</DIV><P></P><FONT SIZE="-1">
+together with Gauss' Divergence Theorem and the properties of the mixed product <!-- MATH
+ $(\underline a\wedge\underline b)\cdot \underline c$
+ -->
+<IMG
+ WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img44.gif"
+ ALT="$ (\underline a\wedge\underline b)\cdot \underline c$"> to obtain
+</FONT><!-- MATH
+ \begin{displaymath}
+\int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - \underline u\cdot \mathop{\rm curl}\underline v  \, dx = \int_{\Omega } \mathrm{div}\, (\underline u \wedge \underline v) \, dx
+	= \int_{\partial \Omega } (\underline u \wedge \underline v)\cdot \underline n \,ds = \int_{\partial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="649" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
+ SRC="img45.gif"
+ ALT="$\displaystyle \int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - ...
+...rtial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+ The extention to a pairing of
+ <!-- MATH
+ $H(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img46.gif"
+ ALT="$ H(\mathop{\rm curl})$"> and <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$"> functions follows with Theorem <A HREF="node1.html#th:density">1</A> by a density argument and is a result of the proof of the
+ Trace Theorem. See [<A
+ HREF="node4.html#Girault-Raviart">4</A>] p.21 for details.
+ 
+</FONT>
+<P>
+<P>
+<DIV><B>T<SMALL>HEOREM</SMALL>  3</B> (Trace Theorem)  &nbsp; 
+For <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">, let <!-- MATH
+ $\underline n$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img47.gif"
+ ALT="$ \underline n$"> denote the outward unit normal to the boundary <!-- MATH
+ $\partial \Omega$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img9.gif"
+ ALT="$ \partial \Omega $">. For <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$">, let <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$"> be as in convention <A HREF="node1.html#def:_tangent">1</A> 
+<BR>
+For <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> the mapping
+<!-- MATH
+ \begin{displaymath}
+\gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img48.gif"
+ ALT="$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t
+$">
+</DIV><P></P>
+and for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$"> the mapping
+<!-- MATH
+ \begin{displaymath}
+\gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img49.gif"
+ ALT="$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n
+$">
+</DIV><P></P>
+is contiuous and linear from <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> to <!-- MATH
+ $[H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$
+ -->
+<IMG
+ WIDTH="97" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img50.gif"
+ ALT="$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$">.</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Note, that the trace of a <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-function is only defined in tangential direction. Its trace is in the dual space
+ of traces of <!-- MATH
+ $[H^1(\Omega )]^{\tilde{d}}$
+ -->
+<IMG
+ WIDTH="74" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img51.gif"
+ ALT="$ [H^1(\Omega )]^{\tilde{d}}$"> functions. Recall that traces of such functions
+are defined in every direction and are functions in <!-- MATH
+ $[H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$
+ -->
+<IMG
+ WIDTH="87" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img52.gif"
+ ALT="$ [H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$">.
+<BR></FONT>
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+The proof of the trace theorem follows from Green's formula stated in theorem <A HREF="node1.html#eq:_PI">2</A> applied to smooth functions
+and then by density arguments. See [<A
+ HREF="node4.html#Girault-Raviart">4</A>] p.21 for details.
+
+</FONT>
+<P>
+<FONT SIZE="-1">Due to the Trace Theorem it makes sense to define a space of <!-- MATH
+ $H(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img46.gif"
+ ALT="$ H(\mathop{\rm curl})$">-functions with vanishing tangential components on the boundary.
+ </FONT><P>
+<DIV><B>D<SMALL>EFINITION</SMALL>  2</B> &nbsp; 
+<!-- MATH
+ \begin{displaymath}
+H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mathop{\rm curl};\Omega ): \quad \underline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="406" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img53.gif"
+ ALT="$\displaystyle H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mat...
+...nderline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\}
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL>  2</B> &nbsp; 
+For <IMG
+ WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ d=2,3$">, <!-- MATH
+ $[\mathcal{D}(\Omega )]^d$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img54.gif"
+ ALT="$ [\mathcal{D}(\Omega )]^d$"> is dense in <!-- MATH
+ $H_0(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img55.gif"
+ ALT="$ H_0(\mathop{\rm curl};\Omega )$">.
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">A consequence of Green's formula is the following important regularity property of <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-functions:
+</FONT>
+<P>
+<P>
+<DIV><A NAME="prop:_no_jumps"><B>P<SMALL>ROPOSITION</SMALL>  1</B></A> &nbsp; 
+Let <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$"> and <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> be two polygonal (resp. polyhedral) Lipschitz domains in <!-- MATH
+ $\mathbb{R}^d$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img58.gif"
+ ALT="$ \mathbb{R}^d$">, with a common edge (resp. common edge or face)
+  <!-- MATH
+ $e = \partial K_-\cap\partial K_+ \neq \emptyset$
+ -->
+<IMG
+ WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img59.gif"
+ ALT="$ e = \partial K_-\cap\partial K_+ \neq \emptyset$"> and
+ denote by <!-- MATH
+ $\Omega = \partial K_-\cup\partial K_+$
+ -->
+<IMG
+ WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img60.gif"
+ ALT="$ \Omega = \partial K_-\cup\partial K_+$"> their union. A function <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img61.gif"
+ ALT="$ v$"> is in <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> if and only if the restricion <IMG
+ WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img62.gif"
+ ALT="$ v_-$"> of <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img61.gif"
+ ALT="$ v$"> to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$"> 
+ is in <!-- MATH
+ $H(\mathop{\rm curl}; K_-)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img63.gif"
+ ALT="$ H(\mathop{\rm curl}; K_-)$">, the restricion <IMG
+ WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img64.gif"
+ ALT="$ v_+$"> of <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img61.gif"
+ ALT="$ v$"> to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> is in <!-- MATH
+ $H(\mathop{\rm curl}; K_+)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img65.gif"
+ ALT="$ H(\mathop{\rm curl}; K_+)$"> <I>and</I> the <I>tangential</I> jump over <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> vanishes: <!-- MATH
+ $v_-\wedge n_- + v_+\wedge n_+ = 0$
+ -->
+<IMG
+ WIDTH="181" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img67.gif"
+ ALT="$ v_-\wedge n_- + v_+\wedge n_+ = 0$"> on <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">.
+ </DIV><P></P>
+<FONT SIZE="-1">
+ </FONT><FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ The proposition follows from choosing an appropriate test function and integrating by parts (global and local).
+ In order to <I>localise</I> the result of the Trace Theorem, we must choose a testfunction from the space <!-- MATH
+ $H^{\frac{1}{2}}_{00}(e)$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
+ SRC="img68.gif"
+ ALT="$ H^{\frac{1}{2}}_{00}(e)$">. These
+ functions vanish at the endpoints of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> and can therefore be extended by zero to a  <!-- MATH
+ $H^{\frac{1}{2}}(\partial \Omega )$
+ -->
+<IMG
+ WIDTH="69" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img69.gif"
+ ALT="$ H^{\frac{1}{2}}(\partial \Omega )$">-function. From the comparison
+ of local (on <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$"> and <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> separately) and global (on <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $">) integration by parts it followas then that the tangential jump vanishes in the
+ dual space of <!-- MATH
+ $H^{\frac{1}{2}}_{00}(e)$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
+ SRC="img68.gif"
+ ALT="$ H^{\frac{1}{2}}_{00}(e)$">. By densitiy properties of <!-- MATH
+ $H^{\frac{1}{2}}_{00}(e)$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
+ SRC="img68.gif"
+ ALT="$ H^{\frac{1}{2}}_{00}(e)$"> it follows that the tangential traces vanish in
+ the "correct space" as well. The "correct space" would be <!-- MATH
+ $H^{-\frac{1}{2}}(e)$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img70.gif"
+ ALT="$ H^{-\frac{1}{2}}(e)$"> if we have no further regularity of <!-- MATH
+ $\underline v_-$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img71.gif"
+ ALT="$ \underline v_-$"> and <!-- MATH
+ $\underline v_+$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img72.gif"
+ ALT="$ \underline v_+$">, and it
+ would be <IMG
+ WIDTH="47" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img73.gif"
+ ALT="$ L^2(e)$"> if <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$"> is elementwise in <IMG
+ WIDTH="30" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img74.gif"
+ ALT="$ H^1$"> (e.&nbsp;g.&nbsp; for piecewise polynomial <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">).
+ 
+</FONT>
+<P>
+
+<H2><A NAME="SECTION00013000000000000000">
+1.3 Variational formulation of the model problem</A>
+</H2><FONT SIZE="-1">
+In the previous sections we introduced the space <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">, an integration-by-parts formula and the notion of trace for an
+<!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-function. In this framework, the variational formulation of the model problem (<A HREF="node1.html#eq:model_problem">1</A>) reads:
+</FONT><DL COMPACT>
+<DT></DT>
+<DD>Find <!-- MATH
+ $\underline u \in H_0(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img75.gif"
+ ALT="$ \underline u \in H_0(\mathop{\rm curl};\Omega )$"> such that for all test functions <!-- MATH
+ $\underline v \in H_0(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="122" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img76.gif"
+ ALT="$ \underline v \in H_0(\mathop{\rm curl};\Omega )$"> holds
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_varform"></A><!-- MATH
+ \begin{equation}
+\int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx =  \int_{\Omega } \underline f\, \cdot \underline v\,dx
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="378" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img77.gif"
+ ALT="$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\...
+..., \cdot \underline v\,dx = \int_{\Omega } \underline f\, \cdot \underline v\,dx$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(3)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+</DD>
+</DL><FONT SIZE="-1">
+With our assumptions on the data, the forms
+</FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+	a(\underline u,\underline v) &:= \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx \\
+	l(\underline v) & := \int_{\Omega } \underline f\, \cdot \underline v\,dx
+\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="454" HEIGHT="92" BORDER="0"
+ SRC="img78.gif"
+ ALT="\begin{displaymath}\begin{split}a(\underline u,\underline v) &amp;:= \int_{\Omega } ...
+...int_{\Omega } \underline f\, \cdot \underline v\,dx \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+are continuous and the bilinear form <!-- MATH
+ $a(\cdot,\cdot)$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img79.gif"
+ ALT="$ a(\cdot,\cdot)$"> is coercive on 
+<!-- MATH
+ $H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="198" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img80.gif"
+ ALT="$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$">. By the Lax-Milgram lemma it follows, that there exists a unique solution 
+<!-- MATH
+ $\underline u \in H_0(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img75.gif"
+ ALT="$ \underline u \in H_0(\mathop{\rm curl};\Omega )$"> of (<A HREF="node1.html#eq:_varform">3</A>).
+</FONT>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
diff --git a/deal.II/doc/reports/nedelec/node2.html b/deal.II/doc/reports/nedelec/node2.html
new file mode 100644
index 0000000000..247c296367
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/node2.html
@@ -0,0 +1,4050 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>2 N&#233;d&#233;lec's elements of first type for </TITLE>
+<META NAME="description" CONTENT="2 N&#233;d&#233;lec's elements of first type for ">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
+
+<UL>
+<LI><A NAME="tex2html59"
+  HREF="#SECTION00021000000000000000">2.1 Construction of N&#233;d&#233;lec elements on tetrahedral grids</A>
+<UL>
+<LI><A NAME="tex2html60"
+  HREF="#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html61"
+  HREF="#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html62"
+  HREF="#SECTION00021300000000000000">2.1.3 Piola transformation</A>
+<LI><A NAME="tex2html63"
+  HREF="#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
+<LI><A NAME="tex2html64"
+  HREF="#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html65"
+  HREF="#SECTION00022000000000000000">2.2 N&#233;d&#233;lec Elements on affine quadrilateral or hexahedral grids</A>
+<UL>
+<LI><A NAME="tex2html66"
+  HREF="#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html67"
+  HREF="#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html68"
+  HREF="#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html69"
+  HREF="#SECTION00023000000000000000">2.3 Construction of N&#233;d&#233;lec elements on bi- or trilinear elements</A>
+<UL>
+<LI><A NAME="tex2html70"
+  HREF="#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
+<LI><A NAME="tex2html71"
+  HREF="#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html72"
+  HREF="#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
+<LI><A NAME="tex2html73"
+  HREF="#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
+</UL>
+<!--End of Table of Child-Links-->
+<HR>
+
+<H1><A NAME="SECTION00020000000000000000">
+2 <FONT SIZE="+1">N&#233;d&#233;lec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+</H1><FONT SIZE="-1">
+	In this section we will present present <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming vector-valued finite elements, 
+	the N&#233;d&#233;lec elements of first type (cf.&nbsp;[<A
+ HREF="node4.html#Ned1">8</A>]), which can be used to discretize the variational problem (<A HREF="node1.html#eq:_varform">3</A>). 
+<BR></FONT>
+<P>
+<FONT SIZE="-1">In order to define a finite element we must specify
+	</FONT><DL>
+<DT><STRONG>the geometry</STRONG></DT>
+<DD>We choose a reference element <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> and a change of variables <!-- MATH
+ $F_K(\hat{x})$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img82.gif"
+ ALT="$ F_K(\hat{x})$">, the element map.
+		 We set <!-- MATH
+ $K = F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$">.
+		
+</DD>
+<DT><STRONG>a function space</STRONG></DT>
+<DD>We need a <I>finite dimensional</I> function space <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$">, typically a space of polynomials, on the reference
+		cell, plus a transformation of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> to a function space <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img85.gif"
+ ALT="$ R_K$"> on a general cell <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">.
+		
+</DD>
+<DT><STRONG>dofs</STRONG></DT>
+<DD>We have to define a set of dofs <!-- MATH
+ $\mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$
+ -->
+<IMG
+ WIDTH="119" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img87.gif"
+ ALT="$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$">. These are linear functionals on 
+		<IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> and <!-- MATH
+ $N < \infty$
+ -->
+<IMG
+ WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img88.gif"
+ ALT="$ N &lt; \infty$"> is the dimension of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$">. <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> should be <I>unisolvent</I>, that is, the dofs <!-- MATH
+ $\alpha_i(\cdot)$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img90.gif"
+ ALT="$ \alpha_i(\cdot)$">
+		are linearly independent.
+	
+</DD>
+</DL>
+<P>
+<FONT SIZE="-1">First, we observe that for a conforming discretization of (<A HREF="node1.html#eq:_varform">3</A>) we cannot take vector-valued finite elements
+	that are build by taking the standard nodal finite element spaces of globally continuous functions for each vector component. 
+	For <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-functions, the only continuity condition is the continuity of the tangential component over cell boundaries. 
+	This fact will motivate the choice of appropriate degrees of freedom (abbreviated by dofs in the following).
+<BR>
+We will give an outline of the construction of the finite element spaces described by N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned1">8</A>]. In literature, they are also referred to as
+ 	<I>N&#233;d&#233;lec's elements of first type</I>.
+<BR></FONT>
+<P>
+
+<H2><A NAME="SECTION00021000000000000000">
+2.1 Construction of N&#233;d&#233;lec elements on tetrahedral grids</A>
+</H2>
+<P>
+<FONT SIZE="-1">In this section, we denote by <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> the standard triangular or tetrahedral reference element.
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00021100000000000000">
+2.1.1 Polynomial spaces on the reference element</A>
+</H3><FONT SIZE="-1">
+			In [<A
+ HREF="node4.html#Ned1">8</A>], N&#233;d&#233;lec introduces the function spaces <!-- MATH
+ $\hat{R} = \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img91.gif"
+ ALT="$ \hat{R} = \mathcal{R}^k$">, on which his finite element will be based. These spaces are subject to this
+			section.
+			For more details, consult [<A
+ HREF="node4.html#Ned1">8</A>].
+<BR>
+We denote by <!-- MATH
+ $\mathbb{P}_k(\hat{\Sigma})$
+ -->
+<IMG
+ WIDTH="51" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img92.gif"
+ ALT="$ \mathbb{P}_k(\hat{\Sigma})$"> the space of polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> on <!-- MATH
+ $\hat{\Sigma}$
+ -->
+<IMG
+ WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img94.gif"
+ ALT="$ \hat{\Sigma}$">, where <!-- MATH
+ $\hat{\Sigma}$
+ -->
+<IMG
+ WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img94.gif"
+ ALT="$ \hat{\Sigma}$"> is an edge, a
+			face of or the reference element itself.. The space <!-- MATH
+ $\tilde{\mathbb{P}}_k$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img95.gif"
+ ALT="$ \tilde{\mathbb{P}}_k$"> of homogeneous 
+			polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> is the span of monomials of total degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img96.gif"
+ ALT="$ d$"> variables on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. 
+</FONT>
+<P>
+<P>
+<DIV><B>D<SMALL>EFINITION</SMALL>  3</B> &nbsp; 
+We define the auxiliary space
+				<P></P>
+<DIV ALIGN="CENTER"><A NAME="def:_space_Sk"></A><!-- MATH
+ \begin{equation}
+\mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d : \underline p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="317" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
+ SRC="img97.gif"
+ ALT="$\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d :...
+...ine p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(4)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+with <!-- MATH
+ $\hat{x} \in \hat{K}$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img98.gif"
+ ALT="$ \hat{x} \in \hat{K}$">.
+<BR>
+The dimension of this space is <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in the case <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> and <IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img99.gif"
+ ALT="$ k(k+2)$"> for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">.
+			</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">N&#233;d&#233;lec's first family of <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite elements is based on the polynomial spaces
+			</FONT><P>
+<DIV><B>D<SMALL>EFINITION</SMALL>  4</B> &nbsp; 
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="def:_space_Rk"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="196" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img100.gif"
+ ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(5)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+These spaces have dimension 
+				<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+					\mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \textrm{for} \quad d=2\,, \\
+					\mathrm{dim} (\mathcal{R}^k) &= \frac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,.
+				\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="446" HEIGHT="72" BORDER="0"
+ SRC="img101.gif"
+ ALT="\begin{displaymath}\begin{split}\mathrm{dim} (\mathcal{R}^k) &amp;= k(k+2) \qquad \t...
+...ac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">In the two-dimensional case, an equivalent characterization of the space <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> is
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_spaces_2d"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1
+				\end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="293" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img103.gif"
+ ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus ...
+..._{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\  -\hat{x}_1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(6)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+			This can be seen by noting that for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> 
+			</FONT><!-- MATH
+ \begin{displaymath}
+\tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \subseteq \mathcal{S}^k
+				%%\left\{\v p\quad \big|\quad \v p = \tilde{p} \left(\begin{array}{cc} x_2 \\-x_1 \end{array}\right)\,,\, \tilde{p}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="164" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img104.gif"
+ ALT="$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\  ...
+...ilde{p} \left(\begin{array}{cc} x_2 \\ -x_1 \end{array}\right)\,,\, \tilde{p}
+$">
+</DIV><P></P><FONT SIZE="-1"> 
+			obviously holds.
+			Moreover, the dimension of the space <!-- MATH
+ $\tilde{\mathbb{P}}_{k-1}$
+ -->
+<IMG
+ WIDTH="43" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img105.gif"
+ ALT="$ \tilde{\mathbb{P}}_{k-1}$"> of homogeneous polynomials of degree <IMG
+ WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img106.gif"
+ ALT="$ k-1$"> in two variables is <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> and this is
+			also the dimension <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">. This proves the stated equivalent representation of the space <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">.
+<BR>			
+<BR>
+We illustrate these definitions with some examples. We start with the case <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> and consider the spaces of polynomials of degree
+			<IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$"> and <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img109.gif"
+ ALT="$ k=2$">:
+</FONT>
+<P>
+<P>
+<DIV><B>E<SMALL>XAMPLE</SMALL>  1</B> &nbsp; 
+
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_R1"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^1 = \left\langle
+								\left(\begin{array}{cc} 1 \\0 \end{array}\right)\,_,
+								\left(\begin{array}{cc} 0 \\1 \end{array}\right)\,_,
+								\left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \right\rangle
+					
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="278" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img110.gif"
+ ALT="$\displaystyle \mathcal{R}^1 = \left\langle \left(\begin{array}{cc} 1 \\  0 \end...
+...eft(\begin{array}{cc} \hat{x}_2 \\  -\hat{x}_1 \end{array}\right) \right\rangle$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(7)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+<!-- MATH
+ \begin{displaymath}
+\mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
+							\left\langle 
+								\left(\begin{array}{cc} \hat{x}_1\,\hat{x}_2 \\-{\hat{x}_1}^2 \end{array}\right)\,_,
+								\left(\begin{array}{cc} {\hat{x}_2}^2 \\-\hat{x}_1\,\hat{x}_2 \end{array}\right)
+							\right\rangle					
+					
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img111.gif"
+ ALT="$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
+\...
+...{cc} {\hat{x}_2}^2 \\  -\hat{x}_1\,\hat{x}_2 \end{array}\right)
+\right\rangle
+$">
+</DIV><P></P>
+
+<P>
+</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">To illustrate a case for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">, we consider the lowest polynomial degree <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">:
+			</FONT><P>
+<DIV><B>E<SMALL>XAMPLE</SMALL>  2</B> &nbsp; 
+We have to specify a basis for <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$">:
+<BR>
+Let <!-- MATH
+ $\underline p$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img113.gif"
+ ALT="$ \underline p$"> be a polynomial in <!-- MATH
+ $(\mathbb{P}_{1}(\hat{K}))^3$
+ -->
+<IMG
+ WIDTH="75" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img114.gif"
+ ALT="$ (\mathbb{P}_{1}(\hat{K}))^3$"> with componentwise representation
+				<!-- MATH
+ \begin{displaymath}
+p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="234" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img115.gif"
+ ALT="$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
+$">
+</DIV><P></P>
+The condition for <!-- MATH
+ $\underline p$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img113.gif"
+ ALT="$ \underline p$"> being in <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$"> is 
+				<!-- MATH
+ \begin{displaymath}
+\underline p \cdot \hat{\underline x} =
+						\sum_{i=1}^3 a_{ii}\hat{x}_i^2 + \sum_{\substack{i,j=1 \\j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
+				
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="327" HEIGHT="86" ALIGN="MIDDLE" BORDER="0"
+ SRC="img116.gif"
+ ALT="$\displaystyle \underline p \cdot \hat{\underline x} =
+\sum_{i=1}^3 a_{ii}\hat{...
+...um_{\substack{i,j=1 \\  j&gt;i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
+$">
+</DIV><P></P>
+This leads to the condition on the coefficients of a polynomial in <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$">:
+				<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+					&a_{11}=a_{22}=a_{33} = 0 \\
+					&a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,.
+				\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="439" HEIGHT="55" BORDER="0"
+ SRC="img117.gif"
+ ALT="\begin{displaymath}\begin{split}&amp;a_{11}=a_{22}=a_{33} = 0 \\  &amp;a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+With the basis of <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$"> which is obtained by choosing <!-- MATH
+ $a_{ij} = 1$
+ -->
+<IMG
+ WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img118.gif"
+ ALT="$ a_{ij} = 1$">, <IMG
+ WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img119.gif"
+ ALT="$ i=1,2,3$">, <IMG
+ WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img120.gif"
+ ALT="$ j&gt;i$"> and setting all the other 
+				coefficients to zero, we get
+				<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus
+													 \left\langle 
+													 	\left(\begin{array}{ccc} 0 \\\hat{x}_3 \\\hat{x}_2 \end{array}\right)\,_,\,
+														\left(\begin{array}{ccc} \hat{x}_3 \\0 \\\hat{x}_1 \end{array}\right)\,_,\,
+														\left(\begin{array}{ccc} \hat{x}_2 \\\hat{x}_1 \\0 \end{array}\right)
+														\right\rangle
+				
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="402" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img121.gif"
+ ALT="$\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus \l...
+...egin{array}{ccc} \hat{x}_2 \\  \hat{x}_1 \\  0 \end{array}\right) \right\rangle$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">We remark at this point that the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> do not span the whole <!-- MATH
+ $(\mathbb{P}_{k}(\hat{K}) )^d$
+ -->
+<IMG
+ WIDTH="76" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img122.gif"
+ ALT="$ (\mathbb{P}_{k}(\hat{K}) )^d$">. 
+			An <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM based on full polynomial spaces, the so called 
+			<I>N&#233;d&#233;lec elements of second type</I>, was introduced in 1986 by N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned2">10</A>].
+</FONT>
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL>  3</B> &nbsp; 
+The original, rather technical, representation of the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> is given in Definition 2 in [<A
+ HREF="node4.html#Ned1">8</A>]. 
+				N&#233;d&#233;lec uses this representation in most of his proofs. We will not refer to it here.
+			</DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00021200000000000000"></A>
+		<A NAME="subsubsect:_dofs_on_tria"></A>
+<BR>
+2.1.2 Degrees of freedom on the reference element
+</H3><FONT SIZE="-1">
+			In this section we define the set <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> of dofs, which is a set of linear functionals on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">.
+			</FONT><P>
+<DIV><B>R<SMALL>EMARK</SMALL>  4</B> &nbsp; 
+Recall that the dimension of the spaces of polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img123.gif"
+ ALT="$ n$"> variables is <!-- MATH
+ $n+k+2 \choose n$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img124.gif"
+ ALT="$ n+k+2 \choose n$">. 
+			</DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_dofs_2d"><B>D<SMALL>EFINITION</SMALL>  5</B></A> &nbsp; 
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference triangle and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>. 
+				The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> in the 2d case consists of the linear functionals
+				<DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall \hat{\varphi }
+						\in \mathbb{P}_{k-1}(\hat{e})\,,
+					
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img126.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img128.gif"
+ ALT="$ 3k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi }
+						\in (\mathbb{P}_{k-2}(\hat{K}))^2\,.
+					
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img129.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
+...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img130.gif"
+ ALT="$ k(k-1)$"> of inner dofs.
+
+<P>
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_dofs_3d"><B>D<SMALL>EFINITION</SMALL>  6</B></A> &nbsp; 
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference tetrahedron, <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A> 
+				and <!-- MATH
+ $\hat{\underline n}$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img131.gif"
+ ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face. 
+				The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> in the 3d case consists of the linear functionals
+				<DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall
+						\hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+					
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img126.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of the tetrahedron <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img132.gif"
+ ALT="$ 6k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>face dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \quad
+						\forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,
+					
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="335" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img133.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
+... }\,d\hat{a} \quad \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every face <IMG
+ WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img134.gif"
+ ALT="$ \hat{f}$"> of the tetrahedron <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img135.gif"
+ ALT="$ 4k(k-1)$"> of face dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad
+						\forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.
+					
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img136.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
+...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <!-- MATH
+ $\frac{k(k-1)(k-2)}{2}$
+ -->
+<IMG
+ WIDTH="88" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img137.gif"
+ ALT="$ \frac{k(k-1)(k-2)}{2}$"> of inner dofs.
+				
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">We note that in the case of lowest order elements, i.&nbsp;e. <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">, only edge dofs occur. This is not so for higher order elements. For
+			<IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img109.gif"
+ ALT="$ k=2$"> we additionally have inner dofs in the 2d case and face dofs in the 3d case. For <IMG
+ WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img138.gif"
+ ALT="$ k\leq3$"> we have all types of dofs in both
+			cases.
+<BR>
+We also note that the total number of dofs equals the dimension of the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">, as it should be.
+<BR>
+The representation of the <I>interface</I> dofs, that is edge dofs in 2d, edge and face dofs in 3d, is motivated by the continuity
+			condition on <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-functions stated in proposition <A HREF="node1.html#prop:_no_jumps">1</A>.
+</FONT>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  2</B> &nbsp; 
+The set <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> of dofs befined above is unisolvent on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">. 
+				<!-- MATH
+ $\hat{\underline u}\in \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="61" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img139.gif"
+ ALT="$ \hat{\underline u}\in \mathcal{R}^k$"> is uniquely defined by the moments <!-- MATH
+ $\hat{\alpha}(\hat{\underline u})$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img140.gif"
+ ALT="$ \hat{\alpha}(\hat{\underline u})$">.
+			</DIV><P></P>
+<FONT SIZE="-1">
+			</FONT><FONT SIZE="-1">P<SMALL>ROOF. </SMALL>See [<A
+ HREF="node4.html#Ned1">8</A>], proof of theorem 1 and preceeding lemmas.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="ex:_shape_functions"><B>E<SMALL>XAMPLE</SMALL>  3</B></A> (Reference shape functions of lowest order for N&#233;d&#233;lec elements on triangular meshes)  &nbsp; 
+Let the reference element be the triangle 
+				<!-- MATH
+ $\hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$
+ -->
+<IMG
+ WIDTH="374" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img141.gif"
+ ALT="$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$">.
+				Label the edges couterclockwise startung with <!-- MATH
+ $\hat{e}_0 = \overline{(0,0),(1,0)}$
+ -->
+<IMG
+ WIDTH="131" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img142.gif"
+ ALT="$ \hat{e}_0 = \overline{(0,0),(1,0)}$">. 
+				The tangential vectors to the edges are (oriented counterclockwise)
+				<!-- MATH
+ \begin{displaymath}
+\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
+					\hat{\underline t}_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} -1 \\1 \end{array}\right)\,,\quad
+					\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="398" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img143.gif"
+ ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\  0 \end{array...
+...
+\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\  -1 \end{array}\right)\,.
+$">
+</DIV><P></P>
+The underlying function space for lowest order N&#233;d&#233;lec elements on a triangular mesh is <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$"> from (<A HREF="#eq:_____R1"><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>).
+<BR>
+In the case of <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$"> only egde-dofs occur. On <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> we have dofs of the type
+				<!-- MATH
+ $\int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="202" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img145.gif"
+ ALT="$ \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$">. 
+				More precisely, since <!-- MATH
+ $\varphi \equiv 1$
+ -->
+<IMG
+ WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img146.gif"
+ ALT="$ \varphi \equiv 1$"> is a basis for <!-- MATH
+ $\mathbb{P}_{0}(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img147.gif"
+ ALT="$ \mathbb{P}_{0}(\hat{e}_i)$"> we have the three dofs
+				<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="250" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img148.gif"
+ ALT="$\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+In order to construct a FE-basis <!-- MATH
+ $\hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img149.gif"
+ ALT="$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$"> for <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$"> with respect to these dofs, we require
+				<!-- MATH
+ $\hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$
+ -->
+<IMG
+ WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img150.gif"
+ ALT="$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$">. This leads to a linear system for
+				the coefficients of the <!-- MATH
+ $\hat{\underline N}_i$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img151.gif"
+ ALT="$ \hat{\underline N}_i$"> in a general basis of <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$">. In the case of lowest order elements, it is easy to
+				verify that we have
+				<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\\hat{x} \end{array}\right)\,,\quad
+					\hat{\underline N}_1 = \left(\begin{array}{cc} -\hat{y} \\\hat{x} \end{array}\right)\,,\quad
+					\hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\\hat{x}-1 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="446" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img152.gif"
+ ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\  \hat...
+...ine N}_2 = \left(\begin{array}{cc} -\hat{y} \\  \hat{x}-1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(8)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00021300000000000000"></A>	<A NAME="sec:_Piola_for_triangles"></A>
+<BR>
+2.1.3 Piola transformation
+</H3><FONT SIZE="-1">
+				An affine triangle or tetrahedron <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> is described by the affine element map
+				</FONT><!-- MATH
+ \begin{displaymath}
+K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img153.gif"
+ ALT="$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
+$">
+</DIV><P></P>
+<P>
+<FONT SIZE="-1">In standard <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$">-conforming FEM, the shape functions <IMG
+ WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img154.gif"
+ ALT="$ N_i$"> on a general cell <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> are obtained from the reference shape functions 
+				<IMG
+ WIDTH="26" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img155.gif"
+ ALT="$ \hat{N}_i$"> on the reference element <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> by the pull-back
+				</FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="187" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
+ SRC="img156.gif"
+ ALT="$\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+				In the case of <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming N&#233;d&#233;lec FEM we cannot transforme our shape function in this way. The pull-back of a 
+				<!-- MATH
+ $H(\mathop{\rm curl};\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img157.gif"
+ ALT="$ H(\mathop{\rm curl};\hat{K})$">-function needs not to be in <!-- MATH
+ $H(\mathop{\rm curl}; K)$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img158.gif"
+ ALT="$ H(\mathop{\rm curl}; K)$">. In addition, the pull-back is not an <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">-isomorphism and it
+				does not lead to an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method if prescribing the dofs by definitions <A HREF="node2.html#def:_dofs_2d">5</A> or <A HREF="#def:_dofs_____3d"><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>.
+<BR>
+In N&#233;d&#233;lec's FEM (or, more general, in <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM), the shape functions are transformed by the following covariant 
+				transformation for vector-fields:
+</FONT>
+<P>
+<DL COMPACT>
+<DT></DT>
+<DD>The element shape functions <!-- MATH
+ $\underline N_i(x)$
+ -->
+<IMG
+ WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img159.gif"
+ ALT="$ \underline N_i(x)$"> on the element <!-- MATH
+ $K = F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$"> are obtained 
+					from the reference shape functions by
+					<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Piola"></A><!-- MATH
+ \begin{equation}
+\underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) =  \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="329" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
+ SRC="img160.gif"
+ ALT="$\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(9)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+where <!-- MATH
+ $\hat{D}F_K$
+ -->
+<IMG
+ WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img161.gif"
+ ALT="$ \hat{D}F_K$"> is the jacobian <!-- MATH
+ $\frac{d}{d\hat{x}}F_K(\hat{x})$
+ -->
+<IMG
+ WIDTH="73" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img162.gif"
+ ALT="$ \frac{d}{d\hat{x}}F_K(\hat{x})$"> of the element map.
+				
+</DD>
+</DL>
+<P>
+<FONT SIZE="-1">In literature, an equivalent to this transformation for <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$">-conforming FEM (which in that case is a contravariant map) 
+				is referred to as <I>Piola transformation</I>, cf. [<A
+ HREF="node4.html#Brezzi-Fortin">3</A>] pp.&nbsp;97. 
+<BR>
+Here, we will refer to the transformation (<A HREF="node2.html#eq:_Piola">9</A>) of the vector field also as <I>Piola transformation</I>.
+<BR>
+We note that the gradients of scalar nodal <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$">-conforming finite elements transform according to the Piola
+				transformation (<A HREF="node2.html#eq:_Piola">9</A>).
+</FONT>
+<P>
+<FONT SIZE="-1">In the case of tetrahedral elements and affine element map <!-- MATH
+ $F_K(\hat{x}) = B_K \hat{x} + b_k$
+ -->
+<IMG
+ WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img164.gif"
+ ALT="$ F_K(\hat{x}) = B_K \hat{x} + b_k$">, the jacobian <!-- MATH
+ $\hat{D}F_K$
+ -->
+<IMG
+ WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img161.gif"
+ ALT="$ \hat{D}F_K$"> 
+				is just the constant matrix <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img165.gif"
+ ALT="$ B_K$"> and we have 
+				</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Piola_tri"></A><!-- MATH
+ \begin{equation}
+\underline v(x) = \mathcal{P}_K (\hat{\underline v}) =  B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="279" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img166.gif"
+ ALT="$\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(10)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+
+<H3><A NAME="SECTION00021400000000000000"></A>
+				<A NAME="subsubsect:_2d-curl_on_tria"></A>
+<BR>
+2.1.4 Transformation of the curl in 2d
+</H3><FONT SIZE="-1">
+				For <!-- MATH
+ $\Omega \subset\mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img167.gif"
+ ALT="$ \Omega \subset\mathbb{R}^2$">, we noted in remark <A HREF="node1.html#re:_rotation">1</A> that vector fields in <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> can be represented as
+			    rotated <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$"> vector fields. 
+				Moreover, it is easy to verify that
+				</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_rot._jacobian"></A><!-- MATH
+ \begin{equation}
+B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img168.gif"
+ ALT="$\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(11)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+				where <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img169.gif"
+ ALT="$ R$"> is the rotation matrix from remark <A HREF="node1.html#re:_rotation">1</A>.
+				Therefore, the properties of the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) in the 2d case can be derived directly
+				from the properties 
+				of the <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$">-Piola transformation stated in [<A
+ HREF="node4.html#Brezzi-Fortin">3</A>] pp.&nbsp;97.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="th:_piola_2d"><B>T<SMALL>HEOREM</SMALL>  4</B></A> (Some properties of 2d Piola transformation for affine element map)  &nbsp; 
+Let <!-- MATH
+ $\underline v(x) = \mathcal{P}_K(\hat{\underline v})$
+ -->
+<IMG
+ WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img170.gif"
+ ALT="$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$">, <!-- MATH
+ $\varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$
+ -->
+<IMG
+ WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img171.gif"
+ ALT="$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$">, <!-- MATH
+ $\hat{x} =
+F_K^{-1}(x)$
+ -->
+<IMG
+ WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img172.gif"
+ ALT="$ \hat{x} =
+F_K^{-1}(x)$">, with  affine element map <IMG
+ WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img173.gif"
+ ALT="$ F_K$">.
+					<DL COMPACT>
+<DT>(i)</DT>
+<DD>The gradient <!-- MATH
+ $D\underline v$
+ -->
+<IMG
+ WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img174.gif"
+ ALT="$ D\underline v$"> transforms according to
+						<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_gradient_trafo_2d"></A><!-- MATH
+ \begin{equation}
+D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img175.gif"
+ ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(12)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+						
+</DD>
+<DT>(ii)</DT>
+<DD>The curl transforms according to
+							<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:curl_trafo_2d"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="176" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img176.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(13)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+As a consequence we see that <!-- MATH
+ $H(\mathop{\rm curl};K)$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img158.gif"
+ ALT="$ H(\mathop{\rm curl}; K)$"> is isomorphic to <!-- MATH
+ $H(\mathop{\rm curl};\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img157.gif"
+ ALT="$ H(\mathop{\rm curl};\hat{K})$"> under the Piola transformation
+							(<A HREF="node2.html#eq:_Piola_tri">10</A>).
+					
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+					</FONT><DL COMPACT>
+<DT>(i)</DT>
+<DD>Chain rule
+					
+</DD>
+<DT>(ii)</DT>
+<DD>We use that the 2d <!-- MATH
+ $\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img19.gif"
+ ALT="$ \mathop{\rm curl}$"> operator is just the trace of the rotated jacobian <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img177.gif"
+ ALT="$ R\,Dv$">. By remark <A HREF="node2.html#eq:_rot._jacobian">11</A>,
+						we can replace <IMG
+ WIDTH="42" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img178.gif"
+ ALT="$ B_K^{-T}$"> and we get that <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img177.gif"
+ ALT="$ R\,Dv$"> is affine-equivalent to  <!-- MATH
+ $\det B_K^{-1}\,R\,\hat{D}\hat{v}$
+ -->
+<IMG
+ WIDTH="109" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img179.gif"
+ ALT="$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$">,
+						which proves (ii).
+					
+</DD>
+</DL><FONT SIZE="-1">
+				
+</FONT>
+<P>
+<P>
+<DIV><B>C<SMALL>OROLLARY</SMALL>  1</B> &nbsp; 
+From (ii) in theorem <A HREF="node2.html#th:_piola_2d">4</A> we deduce
+					<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="240" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img180.gif"
+ ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\h...
+...}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+and we have, together with (ii) from theorem <A HREF="node2.html#th:_piola_2d">4</A>
+					<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underline u \, dx = | B_K |^{-1}\,\int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\,
+						\widehat{\mathop{\rm curl}}\hat{\underline u}  \,d\hat{x}\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="362" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img181.gif"
+ ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underl...
+...hat{\underline v}\, \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<P>
+
+<H3><A NAME="SECTION00021500000000000000"></A>
+<A NAME="subsubsect:_3d-curl_on_tria"></A>
+<BR>
+2.1.5 Transformation of the curl in 3d
+</H3><FONT SIZE="-1">
+	In three dimensions, we cannot identify the curl-operator with the rotated
+	gradient or with the divergence of a rotated vector field. We cannot, as in 2d, derive a transformation formula for the curl from the
+	transformatin formula of the divergence.
+<BR>
+By the chain rule, we obtain the transformation of the gradient of a vector field <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">, defined by the Piola transformation (<A HREF="#eq:__Piola_tri"><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>) of a reference field <!-- MATH
+ $\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img182.gif"
+ ALT="$ \hat{\underline v}$">:
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_trafo_grad_3d"></A><!-- MATH
+ \begin{equation}
+D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img175.gif"
+ ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(14)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We introduce the skew symmetric matrix <!-- MATH
+ $\mathop{\rm Curl}v$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img183.gif"
+ ALT="$ \mathop{\rm Curl}v$"> as 
+	 </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Curl"></A><!-- MATH
+ \begin{equation}
+\left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="183" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img184.gif"
+ ALT="$\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(15)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We see that <!-- MATH
+ $\mathop{\rm Curl}v = D\underline v^T - D\underline v$
+ -->
+<IMG
+ WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img185.gif"
+ ALT="$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$"> and therefore by (<A HREF="node2.html#eq:_trafo_grad_3d">14</A>)
+	 </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Curl_trafo"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="192" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img186.gif"
+ ALT="$\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(16)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  3</B> (Transformation of the curl in 3d)  &nbsp; 
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference tetrahedron and <!-- MATH
+ $K=F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$"> an affine image of it.
+	 	The curl of a vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the Piola transformation of a reference field <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$">
+		transforms according to
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl_trafo_3d"></A><!-- MATH
+ \begin{equation}
+\left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="307" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img189.gif"
+ ALT="$\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(17)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We obtain the matrix <!-- MATH
+ $\mathrm{M_i}$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img190.gif"
+ ALT="$ \mathrm{M_i}$"> by replacing i-th column of the (constant) jacobian <!-- MATH
+ $D(F_K^{-1}) = B_K^{-1}$
+ -->
+<IMG
+ WIDTH="122" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img191.gif"
+ ALT="$ D(F_K^{-1}) = B_K^{-1}$"> 
+		by the vector <!-- MATH
+ $(\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$
+ -->
+<IMG
+ WIDTH="133" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img192.gif"
+ ALT="$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$">:
+		<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll}
+																(\widehat{\mathop{\rm curl}}\,\hat{v} \circ F_K^{-1})_k (x) & \textrm{if} \quad l=i \\
+																(B_K^{-1})_{kl} & \textrm{if} \quad l\neq i
+												\end{array} \right.
+		
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="351" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
+ SRC="img193.gif"
+ ALT="$\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll} (\...
+...} \quad l=i \\  (B_K^{-1})_{kl} &amp; \textrm{if} \quad l\neq i \end{array} \right.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+		(Note: an alternative, equivalent, transformation formula for the curl in 3d is given in proposition <A HREF="node2.html#prop:_Demko_curl">4</A>).
+	 </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+	 	It holds
+		</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl2Curl"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\mathop{\rm Curl}v})_{23} \\({\mathop{\rm Curl}v})_{31}  \\({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="202" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img194.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\math...
+...\mathop{\rm Curl}v})_{31} \\  ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(18)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	 	We demonstrate the statement of the proposition for the first component of the curl, which is <!-- MATH
+ $(\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$
+ -->
+<IMG
+ WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img195.gif"
+ ALT="$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$">. 
+		Using the transformation (<A HREF="node2.html#eq:_Curl_trafo">16</A>), implicit summation over equal indices and the 
+		abbreviation <!-- MATH
+ $b_{ij} := (B_K^{-1})_{ij}$
+ -->
+<IMG
+ WIDTH="111" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img196.gif"
+ ALT="$ b_{ij} := (B_K^{-1})_{ij}$">, we have
+		</FONT><!-- MATH
+ \begin{displaymath}
+({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="223" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img197.gif"
+ ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
+$">
+</DIV><P></P><FONT SIZE="-1">
+		Writing this out and recalling that <!-- MATH
+ $\mathop{\rm Curl}v$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img183.gif"
+ ALT="$ \mathop{\rm Curl}v$"> is skew symmetric, yields
+		</FONT><!-- MATH
+ \begin{displaymath}
+({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\widehat{\mathop{\rm Curl}} \,\hat{v})_{12}
+							-(b_{12}b_{33} - b_{32}b_{13})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{31}
+							+(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="739" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img198.gif"
+ ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\wide...
+...
++(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+		and with (<A HREF="node2.html#eq:_curl2Curl">18</A>) this is equal to the determinant of 
+		</FONT><!-- MATH
+ \begin{displaymath}
+\mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm curl}}\, v)_1 & b_{12} & b_{13} \\
+									 (\widehat{\mathop{\rm curl}}\, v)_2 & b_{22} & b_{23} \\
+									 (\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33}
+							\end{array}\right) \,.
+		
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="253" HEIGHT="93" ALIGN="MIDDLE" BORDER="0"
+ SRC="img199.gif"
+ ALT="$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm cu...
+...
+(\widehat{\mathop{\rm curl}}\, v)_3 &amp; b_{32} &amp; b_{33}
+\end{array}\right) \,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+		The proof for the other components follows analogously.
+	
+</FONT>
+<P>
+<FONT SIZE="-1">In the next proposition, we state an alternative, equivalent, formula for the transformation of the curl 
+	(e.&nbsp;g.&nbsp;used by Demkovicz in [<A
+ HREF="node4.html#Demko3d">12</A>]) 
+	</FONT><P>
+<DIV><A NAME="prop:_Demko_curl"><B>P<SMALL>ROPOSITION</SMALL>  4</B></A> &nbsp; 
+For a vector field <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$"> on the tetrahedron <!-- MATH
+ $K=F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$">, defined by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) of a reference 
+		field <!-- MATH
+ $\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img182.gif"
+ ALT="$ \hat{\underline v}$"> on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">, we have
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_3d_curl_trafo_Demko"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="271" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img200.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(19)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+		The transformation formula (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) can be proven componentwise, and 
+	    we will only carry out the proof for the first vector component <!-- MATH
+ $(\mathop{\rm curl}\underline v)_1$
+ -->
+<IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img201.gif"
+ ALT="$ (\mathop{\rm curl}\underline v)_1$">. The proofs for the other components
+		follow analogously.
+<BR>
+The identity (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) reads for the first vector component
+		</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_1.comp_of_curl_trafo"></A><!-- MATH
+ \begin{equation}
+(\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="334" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img202.gif"
+ ALT="$\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(20)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1"> 
+		Referring to (<A HREF="node2.html#eq:_curl_trafo_3d">17</A>), we show that the right hand side of (<A HREF="node2.html#eq:_1.comp_of_curl_trafo">20</A>) equals 
+		<!-- MATH
+ $\det \mathrm{M_1}$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img203.gif"
+ ALT="$ \det \mathrm{M_1}$">.
+		For this, we expand <!-- MATH
+ $\det  \mathrm{M_1}$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img203.gif"
+ ALT="$ \det \mathrm{M_1}$"> to
+		</FONT><!-- MATH
+ \begin{displaymath}
+\det  \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det\mathcal{B}^{inv}_{11}
+								-(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det\mathcal{B}^{inv}_{21}
+								+(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
+		
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="505" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img204.gif"
+ ALT="$\displaystyle \det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underlin...
+...ehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+		where <!-- MATH
+ $\mathcal{B}^{inv}_{ij}$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img205.gif"
+ ALT="$ \mathcal{B}^{inv}_{ij}$"> is the <!-- MATH
+ $2 \times 2$
+ -->
+<IMG
+ WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img206.gif"
+ ALT="$ 2 \times 2$">-matrix arising from <IMG
+ WIDTH="39" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img207.gif"
+ ALT="$ B_K^{-1}$"> when cancelling its i-th row and its j-th column.
+<BR>
+We recall the formula for the inverse of a matrix <!-- MATH
+ $A \in \mathbb{R}^{3\times 3}$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img208.gif"
+ ALT="$ A \in \mathbb{R}^{3\times 3}$">
+		</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Ainv"></A><!-- MATH
+ \begin{equation}
+(A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="251" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img209.gif"
+ ALT="$\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(21)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1"> 
+		where <!-- MATH
+ $\mathcal{A}_{ij}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img210.gif"
+ ALT="$ \mathcal{A}_{ij}$"> is the <!-- MATH
+ $2 \times 2$
+ -->
+<IMG
+ WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img206.gif"
+ ALT="$ 2 \times 2$">-matrix arising from <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img211.gif"
+ ALT="$ A$"> when cancelling its i-th row and its j-th column.
+<BR>
+Replacing <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img165.gif"
+ ALT="$ B_K$"> in the right hand side of (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) by the expression (<A HREF="node2.html#eq:_Ainv">21</A>)
+		for <!-- MATH
+ $A = B_K^{-1}$
+ -->
+<IMG
+ WIDTH="75" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img212.gif"
+ ALT="$ A = B_K^{-1}$">, we get
+		</FONT><!-- MATH
+ \begin{displaymath}
+\frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \mathcal{B}^{inv}_{j1} (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_j
+			= (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det \mathcal{B}^{inv}_{11}
+			-(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det \mathcal{B}^{inv}_{21}
+			+(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det  \mathrm{M_1}\,.
+		
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="834" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
+ SRC="img213.gif"
+ ALT="$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \math...
+...l}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+	
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00022000000000000000"></A> <A NAME="sec:_parallelo"></A>
+<BR>
+2.2 N&#233;d&#233;lec Elements on affine quadrilateral or hexahedral grids
+</H2><FONT SIZE="-1">
+	We want to present the ingredients for N&#233;d&#233;lec's finite elements of first type on
+	grids consisiting of parallelograms (in 2d) or the respective objects in 3d, so called parallelotops (cf. section 
+	<I>FE built on cubes</I> in [<A
+ HREF="node4.html#Ned1">8</A>]). Such grids consist
+	 of elements <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> that are affine images of the square or cubic reference element <!-- MATH
+ $\hat{C} = [0,1]^d$
+ -->
+<IMG
+ WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img215.gif"
+ ALT="$ \hat{C} = [0,1]^d$">:
+	</FONT><!-- MATH
+ \begin{displaymath}
+C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img216.gif"
+ ALT="$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
+$">
+</DIV><P></P>
+<P>
+
+<H3><A NAME="SECTION00022100000000000000">
+2.2.1 Polynomial spaces on the reference element</A>
+</H3>
+<P>
+<FONT SIZE="-1">In order to introduce the function spaces needed for the construction of N&#233;d&#233;lec's finite elements, 
+	let us define some spaces of vector-valued polynomials
+	</FONT><P>
+<DIV><B>D<SMALL>EFINITION</SMALL>  7</B> &nbsp; 
+<!-- MATH
+ $\mathcal{Q}_{l,m}$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img217.gif"
+ ALT="$ \mathcal{Q}_{l,m}$"> are the spaces of polynomials on the reference square <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> with maximal degree <IMG
+ WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img219.gif"
+ ALT="$ l$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img220.gif"
+ ALT="$ \hat{x}_1$"> and 
+		<IMG
+ WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img221.gif"
+ ALT="$ m$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img222.gif"
+ ALT="$ \hat{x}_2$">.
+<BR>		<!-- MATH
+ $\mathcal{Q}_{l,m,n}$
+ -->
+<IMG
+ WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img223.gif"
+ ALT="$ \mathcal{Q}_{l,m,n}$"> are the spaces of polynomials on the reference cube <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> with maximal degree <IMG
+ WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img219.gif"
+ ALT="$ l$"> 
+		in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img220.gif"
+ ALT="$ \hat{x}_1$">, <IMG
+ WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img221.gif"
+ ALT="$ m$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img222.gif"
+ ALT="$ \hat{x}_2$"> and <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img123.gif"
+ ALT="$ n$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img224.gif"
+ ALT="$ \hat{x}_3$">.
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">The spaces <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> for the reference shape functions now are in 2d
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_2d_spaces_for_quads"></A><!-- MATH
+ \begin{equation}
+\mathcal{P}^k = \left\{ \hat{\underline u} =
+			\left(\begin{array}{cc} \hat{u}_1 \\\hat{u}_2 \end{array}\right): \quad 
+						 \hat{u}_1 \in \mathcal{Q}_{k-1,k}\,,
+								\hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,
+	
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="402" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img225.gif"
+ ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
+...t{u}_1 \in \mathcal{Q}_{k-1,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(22)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1"> 
+</FONT>
+<P>
+<FONT SIZE="-1">and in 3d
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_3d_spaces_for_quads"></A><!-- MATH
+ \begin{equation}
+\mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{ccc}
+					\hat{u}_1 \\\hat{u}_2 \\\hat{u}_3 \end{array}\right):\quad 
+								\hat{u}_1 \in \mathcal{Q}_{k-1,k,k}\,,
+								\hat{u}_2 \in \mathcal{Q}_{k,k-1,k}\,, 
+								\hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.
+	
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="546" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img226.gif"
+ ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
+...}_2 \in \mathcal{Q}_{k,k-1,k}\,, \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(23)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">  
+</FONT>
+<P>
+<FONT SIZE="-1">We renounce an example, since it is quite evident, what these spaces look like for a specific <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$">.
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00022200000000000000"></A>
+<A NAME="subsubsect:_dofs_on_quads"></A>
+<BR>
+2.2.2 Degrees of freedom on the reference element
+</H3><FONT SIZE="-1">
+	We start with the degrees of freedoms on the reference square <!-- MATH
+ $\hat{C}\subset \mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img227.gif"
+ ALT="$ \hat{C}\subset \mathbb{R}^2$">:
+	</FONT><P>
+<DIV><A NAME="def:_quad-dofs_2d"><B>D<SMALL>EFINITION</SMALL>  8</B></A> &nbsp; 
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> denote the reference square and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>. 
+	The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> in the 2d case consists of the linear functionals
+	<DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
+			\forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+		
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img229.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img230.gif"
+ ALT="$ 4k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}}  \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x}\,, \quad
+			\forall\, \hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
+			\quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.
+		
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="566" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img231.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
+... }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img232.gif"
+ ALT="$ 2k(k-1)$"> of inner dofs.
+
+<P>
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_quad-dofs_3d"><B>D<SMALL>EFINITION</SMALL>  9</B></A> &nbsp; 
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> denote the reference cube, <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A> 
+	and <!-- MATH
+ $\hat{\underline n}$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img131.gif"
+ ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face.  
+	The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> in the 3d case consists of the linear functionals
+	<DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
+			\forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+			
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img229.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <IMG
+ WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img233.gif"
+ ALT="$ 12k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>face dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \,,\quad
+			\forall \,\hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
+			\quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="651" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img234.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
+..._{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every face <IMG
+ WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img134.gif"
+ ALT="$ \hat{f}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <!-- MATH
+ $6\cdot 2k(k-1)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img235.gif"
+ ALT="$ 6\cdot 2k(k-1)$"> of face dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}}  \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \,,\quad
+			\forall\, \hat{\underline \varphi } = \left(\begin{array}{ccc} \hat{\varphi }_1 \\\hat{\varphi }_2 \\
+			\hat{\varphi }_3\end{array}\right) \,,\quad\hat{\varphi }_1\in\mathcal{Q}_{k-1,k-2,k-2}\,,\quad\hat{\varphi _2}\in\mathcal{Q}_{k-2,k-1,k-2}\,,
+			\quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.
+		
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="793" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
+ SRC="img236.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
+...athcal{Q}_{k-2,k-1,k-2}\,, \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="86" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img237.gif"
+ ALT="$ 3k(k-1)^2$"> of inner dofs.
+	
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><B>E<SMALL>XAMPLE</SMALL>  4</B> &nbsp; 
+Proceeding the same way as in example <A HREF="node2.html#ex:_shape_functions">3</A> for a triangular reference element, we obtain
+the reference shape functions of lowest
+order on the square <IMG
+ WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img238.gif"
+ ALT="$ [0,1]^2$">. For the unit tangents as in convention <A HREF="node1.html#def:_tangent">1</A>
+<!-- MATH
+ \begin{displaymath}
+\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
+		\hat{\underline t}_1 = \left(\begin{array}{cc} 0 \\1 \end{array}\right)\,,\quad
+		\hat{\underline t}_2 = \left(\begin{array}{cc} -1 \\0 \end{array}\right)\,, \quad
+		\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="487" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img239.gif"
+ ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\  0 \end{array...
+...
+\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\  -1 \end{array}\right)\,,
+$">
+</DIV><P></P>
+they read
+	<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\0 \end{array}\right)\,,\quad
+		\hat{\underline N}_1 = \left(\begin{array}{cc} 0 \\\hat{x} \end{array}\right)\,,\quad
+		\hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\0 \end{array}\right)\,,\quad
+		\hat{\underline N}_3 = \left(\begin{array}{cc} 0 \\\hat{x}-1 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="573" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img240.gif"
+ ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\  0 \e...
+...\underline N}_3 = \left(\begin{array}{cc} 0 \\  \hat{x}-1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(24)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00022300000000000000">
+2.2.3 Transformation of the vector field</A>
+</H3><FONT SIZE="-1">
+Since the elements of the considered grids are still affine images of the reference element, we can use the Piola transformation 
+(<A HREF="node2.html#eq:_Piola_tri">10</A>) to transform vector fields and the results stated in sections 
+<A HREF="node2.html#sec:_Piola_for_triangles">2.1.3</A> - <A HREF="node2.html#subsubsect:_3d-curl_on_tria">2.1.5</A> can be carried over one to one.
+</FONT>
+<H2><A NAME="SECTION00023000000000000000">
+2.3 Construction of N&#233;d&#233;lec elements on bi- or trilinear elements</A>
+</H2><FONT SIZE="-1">
+	We now want to consider grids that are composed of elements that are a bi- resp. trilinear images <!-- MATH
+ $F_C(\hat{C})$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img241.gif"
+ ALT="$ F_C(\hat{C})$"> of the reference 
+	element <!-- MATH
+ $\hat{C} = [0,1]^d$
+ -->
+<IMG
+ WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img215.gif"
+ ALT="$ \hat{C} = [0,1]^d$">.
+ 	The main difference here is, that the jacobian <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> of the element map <IMG
+ WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img243.gif"
+ ALT="$ F_C$"> is not
+ 	constant, and we have to use Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) to
+ 	transform vector fields. 
+<BR></FONT>
+<P>
+
+<H3><A NAME="SECTION00023100000000000000">
+2.3.1 Bilinear elements in 2d</A>
+</H3><FONT SIZE="-1">
+		The polynomial spaces <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine quadrilateral elements.
+<BR>
+A transformed vector field on a general element is now defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) 
+		</FONT><!-- MATH
+ \begin{displaymath}
+\underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="212" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img244.gif"
+ ALT="$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$">
+</DIV><P></P><FONT SIZE="-1"> of a vector field on the reference element. Note that
+		the jacobian <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> is not constant in this case. In contrast to the case of affine elements, the gradient <!-- MATH
+ $D\underline v$
+ -->
+<IMG
+ WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img174.gif"
+ ALT="$ D\underline v$"> does not transform according to 
+		formula (<A HREF="node2.html#eq:_gradient_trafo_2d">12</A>). Non-vanishing second derivatives of <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> appear in the transformation rule
+		for gradients of vector fields.
+		This requires a new approach to express <!-- MATH
+ $\mathop{\rm curl}\underline v$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img245.gif"
+ ALT="$ \mathop{\rm curl}\underline v$"> in terms of
+		<!-- MATH
+ $\widehat{\mathop{\rm curl}}\,\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img246.gif"
+ ALT="$ \widehat{\mathop{\rm curl}}\,\hat{\underline v}$">. 
+		Nevertheless, it can be shown that the curl of a vector field transforms analogously to the case of affine elements.
+</FONT>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  5</B> &nbsp; 
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> be the reference element <IMG
+ WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img238.gif"
+ ALT="$ [0,1]^2$"> and <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> a bilinear image of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. If the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$">
+			transforms according to the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>), then the transformation of the curl obeys
+			<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad  x = F(\hat{x})\,,
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="370" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img247.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+as in the affine case. 
+<BR>		</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+			In this proof, the mapped element <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> will be fixed, so for simplicity we write <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img248.gif"
+ ALT="$ F$"> for <IMG
+ WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img243.gif"
+ ALT="$ F_C$">.
+<BR>
+First note that <!-- MATH
+ $(\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$
+ -->
+<IMG
+ WIDTH="242" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img249.gif"
+ ALT="$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$">. We use the notation <!-- MATH
+ $D(F^{-1})_{ij}(x)=
+\frac{\partial \hat{x}_i}{\partial x_j}(x)$
+ -->
+<IMG
+ WIDTH="174" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img250.gif"
+ ALT="$ D(F^{-1})_{ij}(x)=
+\frac{\partial \hat{x}_i}{\partial x_j}(x)$"> and imlicit summation to rewrite the Piola transformation of the vector field componentwise
+			</FONT><!-- MATH
+ \begin{displaymath}
+v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img251.gif"
+ ALT="$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+			In the case of affine elements, i.&nbsp;e.&nbsp;for constant jacobian, we have
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+			 \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
+			 \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,,
+			\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="398" HEIGHT="90" BORDER="0"
+ SRC="img252.gif"
+ ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &amp;= \frac{\part...
+... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">whereas for non-constant jacobian we have
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+			 \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) +
+			 						  \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
+			 \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) +
+			 						  \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,.
+			\end{split}
+			
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="493" HEIGHT="95" BORDER="0"
+ SRC="img253.gif"
+ ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &amp;= \frac{\part...
+... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We see that in <I>both</I> cases we have 
+			</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1} - \frac{\partial v_1}{\partial x_2} =
+							\frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) 
+							- \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
+			
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="548" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img254.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1}...
+...ial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
+$">
+</DIV><P></P><FONT SIZE="-1">
+			that is, the second derivatives cancel out in the expression for the curl and the curl in the non-affine case transforms equally
+			to the curl in the affine case.
+		
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00023200000000000000">
+2.3.2 Trilinear elements in 3d</A>
+</H3><FONT SIZE="-1">
+		The polynomial spaces <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine hexahedral elements.
+<BR>
+The
+		vector field on a genereal element is defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>).
+<BR>
+The problem of the non-vanishing second derivatives of the jacobian <!-- MATH
+ $D(F_C^{-1})(x)$
+ -->
+<IMG
+ WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img255.gif"
+ ALT="$ D(F_C^{-1})(x)$"> arises again, and we cannot generalize the results
+		from the affine case straight away.
+<BR>
+But analogously to the 2d case, one can check that in in the transformation rule for expressions 
+		<!-- MATH
+ $\frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img256.gif"
+ ALT="$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$">, <!-- MATH
+ $i,j = 1,2,3$
+ -->
+<IMG
+ WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img257.gif"
+ ALT="$ i,j = 1,2,3$">,
+		which define the curl-operator, the terms containing second derivatives vanish. We have therefore again the transformation rule 
+		(<A HREF="node2.html#eq:_Curl_trafo">16</A>) for the skew matrix <!-- MATH
+ $\mathop{\rm Curl}v = Dv^T - Dv$
+ -->
+<IMG
+ WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img258.gif"
+ ALT="$ \mathop{\rm Curl}v = Dv^T - Dv$">:
+		</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,\hat{D}F_C^{-1}) \circ F_C^{-1})(x)
+						= (DF_C^{-1})^T(x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="673" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img259.gif"
+ ALT="$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm C...
+...x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
+$">
+</DIV><P></P>
+<P>
+<FONT SIZE="-1">It follows that the following proposition can be proved analogously to the case of affine elements (replace there <IMG
+ WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img260.gif"
+ ALT="$ B_C$"> by 
+		<!-- MATH
+ $\hat{D}F_C(\hat(x))$
+ -->
+<IMG
+ WIDTH="81" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img261.gif"
+ ALT="$ \hat{D}F_C(\hat(x))$">).	
+		</FONT><P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  6</B> &nbsp; 
+Let the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on a trilinear image <!-- MATH
+ $C = F_C(\hat{C})$
+ -->
+<IMG
+ WIDTH="93" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img262.gif"
+ ALT="$ C = F_C(\hat{C})$"> be defined by the Piola transformation of a 
+			reference field <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. The transformation formula for the curl then reads
+			<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathop{\rm curl}\underline v =
+				\left(\frac{1}{\det \hat{D}F_C}\,\hat{D}F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.
+			
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
+ SRC="img263.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \left(\frac{1}{\det \hat{D}F_C}\,...
+...F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00024000000000000000">
+2.4 Construction of global shape functions</A>
+</H2><FONT SIZE="-1">
+	In the previous sections we have introduced function spaces and degrees of freedom, which, together with the Piola transformation,
+	 will allow us to define an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method.
+	Indeed, in [<A
+ HREF="node4.html#Ned1">8</A>], N&#233;d&#233;lec shows the invariance of the spaces
+	<!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> and <!-- MATH
+ $\mathcal{Q}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img264.gif"
+ ALT="$ \mathcal{Q}^k$"> under Piola transformation of the vector field, as well as the unisolvence of the set of 
+	degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> from sections <A HREF="node2.html#subsubsect:_dofs_on_tria">2.1.2</A> and <A HREF="node2.html#subsubsect:_dofs_on_quads">2.2.2</A>  
+	(for details, see [<A
+ HREF="node4.html#Ned1">8</A>], Section 1.2, Theorem 1 and Section 2, Theorem 5).
+	This leads to the fact that <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions can be defined by mapping elementwise the reference
+	shape functions with the Piola transformation <!-- MATH
+ $\mathcal{P}_K$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img265.gif"
+ ALT="$ \mathcal{P}_K$">.
+	However, we must pay some care to the orientation of an interface on which the moments defining the degrees of freedom are based.
+	For the 2d case, we will illustrate in this section how we must take into account the <I>orientation of an edge</I> 
+	in the definition of the respective element edge shape function, in order to get an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element space 
+	of global shape functions.
+	
+<BR></FONT>
+<P>
+<FONT SIZE="-1">Let <!-- MATH
+ $K=F(\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img266.gif"
+ ALT="$ K=F(\hat{K})$"> be an affine or bilinear image of a reference element, <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> one of its edges and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> the
+	corresponding edge on the reference element.
+<BR>
+Let further <!-- MATH
+ $[0,|e|] \ni s \mapsto \underline x(s) \in e$
+ -->
+<IMG
+ WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img267.gif"
+ ALT="$ [0,\vert e\vert] \ni s \mapsto \underline x(s) \in e$"> and <!-- MATH
+ $[0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$
+ -->
+<IMG
+ WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img268.gif"
+ ALT="$ [0,\vert\hat{e}\vert] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$"> be parametrizations
+	with respect to the arc length of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> respectively. We can assume that these parametrizations endow the edges with a
+	counterclockwise orientation. Then, the unit tangent vectors <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$"> and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> are given by
+	<!-- MATH
+ $\frac{d \underline x}{ds}$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
+ SRC="img269.gif"
+ ALT="$ \frac{d \underline x}{ds}$"> and <!-- MATH
+ $\frac{d \hat{\underline x}}{d\hat{s}}$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
+ SRC="img270.gif"
+ ALT="$ \frac{d \hat{\underline x}}{d\hat{s}}$">.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="lemma:__v_dot_t_-Trafo"><B>L<SMALL>EMMA</SMALL>  1</B></A> &nbsp; 
+Let <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> be a vector field on the reference element and <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> be the corresponding vector field on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the
+		Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>). It then holds
+		<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img271.gif"
+ ALT="$\displaystyle \underline v\cdot \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert} (\hat{\underline v}\cdot \hat{\underline t})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(25)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+where <IMG
+ WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img272.gif"
+ ALT="$ \vert\hat{e}\vert$"> and <IMG
+ WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img273.gif"
+ ALT="$ \vert e\vert$"> denote the length of the edges <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">.
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+		With 
+		</FONT><!-- MATH
+ \begin{displaymath}
+(\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="300" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img274.gif"
+ ALT="$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
+$">
+</DIV><P></P><FONT SIZE="-1">
+		and <!-- MATH
+ $\hat{x}_j = \hat{x}_j(\underline x(s))$
+ -->
+<IMG
+ WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img275.gif"
+ ALT="$ \hat{x}_j = \hat{x}_j(\underline x(s))$"> and <!-- MATH
+ $\hat{x}_j = \hat{x}_j(\hat{s}(s))$
+ -->
+<IMG
+ WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img276.gif"
+ ALT="$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$"> on the edges, we have
+		</FONT><!-- MATH
+ \begin{displaymath}
+\underline v\cdot \underline t= \underline v \cdot \frac{d \underline x}{ds} = \left( \hat{\underline v}_j\frac{\partial \hat{x}_j}{\partial x_i}\right) (x)\frac{dx_i}{ds}
+			= \hat{\underline v}_j \frac{d \hat{x}_j}{ds} = \hat{\underline v}_j \frac{d \hat{x}_j}{d\hat{s}} \frac{d\hat{s}}{ds}
+			= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds} 
+		
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="498" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
+ SRC="img277.gif"
+ ALT="$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \under...
+...t{s}}{ds}
+= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds}
+$">
+</DIV><P></P><FONT SIZE="-1">
+		and with <!-- MATH
+ $\frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img278.gif"
+ ALT="$ \frac{d\hat{s}}{ds}=\frac{\vert\hat{e}\vert}{\vert e\vert}$"> the lemma follows.
+	
+</FONT>
+<P>
+<FONT SIZE="-1">As a consequence, we have
+</FONT>
+<P>
+<P>
+<DIV><A NAME="prop:_Dof_invariance"><B>P<SMALL>ROPOSITION</SMALL>  7</B></A> (Invariance of the edge dofs)  &nbsp; 
+Let the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> be defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) of a reference vector field 
+		<!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. Then, the functionals (<I>edge dofs</I>) 
+		<!-- MATH
+ $\alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$
+ -->
+<IMG
+ WIDTH="178" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img279.gif"
+ ALT="$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$"> are invariant in the sense of
+		<!-- MATH
+ \begin{displaymath}
+\alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline t)\varphi \,ds =
+			\int_{\hat{e}} (\hat{\underline v} \cdot \hat{\underline t}) \hat{\varphi } \, d\hat{s}\, = \hat{\alpha}(\hat{\underline u})\,,
+			\qquad \forall\, \hat{\varphi } \in
+			\mathbb{P}_{k-1}(\hat{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
+		
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="704" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
+ SRC="img280.gif"
+ ALT="$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline...
+...{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Let now <!-- MATH
+ $K_- = F_-(\hat{K})$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img281.gif"
+ ALT="$ K_- = F_-(\hat{K})$"> and <!-- MATH
+ $K_+ = F_+(\hat{K})$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img282.gif"
+ ALT="$ K_+ = F_+(\hat{K})$"> be two neighbouring triangles with common edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">. Let <!-- MATH
+ $\underline N$
+ -->
+<IMG
+ WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img283.gif"
+ ALT="$ \underline N$"> be the global edge shape
+	function that 'lives' on <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">. By <!-- MATH
+ $\underline N_-$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img284.gif"
+ ALT="$ \underline N_-$"> and <!-- MATH
+ $\underline N_+$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img285.gif"
+ ALT="$ \underline N_+$"> we denote the restriction of <!-- MATH
+ $\underline N$
+ -->
+<IMG
+ WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img283.gif"
+ ALT="$ \underline N$"> to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> and <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$"> respectively.
+	Let <!-- MATH
+ $e_+ =F_+(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img286.gif"
+ ALT="$ e_+ =F_+(\hat{e}_i)$"> and <!-- MATH
+ $e_- =F_-(\hat{e}_j)$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img287.gif"
+ ALT="$ e_- =F_-(\hat{e}_j)$">.
+	We write <!-- MATH
+ $\underline t_+$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img288.gif"
+ ALT="$ \underline t_+ $"> for the tangential unit vector to <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">, oriented 
+	counterclockwise with respect to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> and <!-- MATH
+ $\underline t_- = -\underline t_+$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img289.gif"
+ ALT="$ \underline t_- = -\underline t_+$"> 
+	for the respective from <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$">. For line integrals over the edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> we write <!-- MATH
+ $\int_{e_+}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img290.gif"
+ ALT="$ \int_{e_+}$"> if we chose the orientation induced by 
+	<!-- MATH
+ $\underline t_+$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img288.gif"
+ ALT="$ \underline t_+ $"> and <!-- MATH
+ $\int_{e_-}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img291.gif"
+ ALT="$ \int_{e_-}$"> for the orientation of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> induced by <!-- MATH
+ $\underline t_-$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img292.gif"
+ ALT="$ \underline t_-$">. 
+<BR>
+In order to obtain an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method, proposition <A HREF="node1.html#prop:_no_jumps">1</A> tells us that we must ensure the continuity of the
+	tangential components of the global shape functions, that is
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_conformity_of_shape_func"></A><!-- MATH
+ \begin{equation}
+\underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="182" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img293.gif"
+ ALT="$\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(26)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	The following lemma will justify the choice of the moments describing the edge dofs. A consequence of the lemma will be, 
+	that the matching of the local edge dofs <!-- MATH
+ $\alpha^{[K_+]}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img294.gif"
+ ALT="$ \alpha^{[K_+]}$"> and <!-- MATH
+ $\alpha^{[K_-]}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img295.gif"
+ ALT="$ \alpha^{[K_-]}$"> guarantees  
+	the pointwise condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
+	</FONT><P>
+<DIV><A NAME="lemma:_trace"><B>L<SMALL>EMMA</SMALL>  2</B></A> &nbsp; 
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> denote the reference triangle and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> one of its edges, parametrized by <!-- MATH
+ $\hat{e} \ni \hat{x}(s) := \underline a +
+s\, \hat{\underline t}$
+ -->
+<IMG
+ WIDTH="143" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img296.gif"
+ ALT="$ \hat{e} \ni \hat{x}(s) := \underline a +
+s\, \hat{\underline t}$">. 
+		Let <!-- MATH
+ $\hat{\underline p} \in \mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img297.gif"
+ ALT="$ \hat{\underline p} \in \mathcal{S}^k$">, <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$"> as defined in (<A HREF="node2.html#def:_space_Sk">4</A>). It then holds
+		<!-- MATH
+ \begin{displaymath}
+(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="146" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img298.gif"
+ ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+			</FONT><!-- MATH
+ \begin{displaymath}
+\hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad \textrm{for } i=1,2,3:
+				\quad \hat{p}_i(\hat{x}) = \prod_{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
+				\textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
+			
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="545" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img299.gif"
+ ALT="$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad ...
+...{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
+\textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+			Hence, with the parametrization of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> by <!-- MATH
+ $\hat{x}(s)$
+ -->
+<IMG
+ WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img300.gif"
+ ALT="$ \hat{x}(s)$">
+			</FONT><!-- MATH
+ \begin{displaymath}
+\hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
+						\hat{\varphi }_{k-1}(s)\,,
+			
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="390" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img301.gif"
+ ALT="$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
+\hat{\varphi }_{k-1}(s)\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+			with <!-- MATH
+ $\hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$
+ -->
+<IMG
+ WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img302.gif"
+ ALT="$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$">, 
+			and 
+			</FONT><!-- MATH
+ \begin{displaymath}
+(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} = s^k\,\sum_{i=1}^3\hat{t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
+				\hat{\varphi }_{k-1}(s)\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
+ SRC="img303.gif"
+ ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} = s^...
+...t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
+\hat{\varphi }_{k-1}(s)\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+			We observe that the coefficient of <IMG
+ WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img304.gif"
+ ALT="$ s^k$"> is exactly <!-- MATH
+ $\hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img305.gif"
+ ALT="$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$">. By the definition of the space 
+			<!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">, this expression must vanish.
+	
+</FONT>
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL>  5</B> &nbsp; 
+In the case of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> being a quadrilateral, we have <!-- MATH
+ $\hat{R} = \mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img306.gif"
+ ALT="$ \hat{R} = \mathcal{P}^k$">. By the definition of <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> we see
+		immediately that here also <!-- MATH
+ $(\hat{\underline v}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img307.gif"
+ ALT="$ (\hat{\underline v}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$">.
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">The next proposition tells us how exactly to define element shape functions on a mapped element <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> in order to get 
+	<!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions.
+	</FONT><P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  8</B> &nbsp; 
+Condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>) is satiesfied, if we define the element shape functions <!-- MATH
+ $\underline N_+$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img285.gif"
+ ALT="$ \underline N_+$"> and <!-- MATH
+ $\underline N_-$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img284.gif"
+ ALT="$ \underline N_-$"> 
+		by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) and take into account the orientation of the edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">: 
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_trafo_shape_functions"></A><!-- MATH
+ \begin{equation}
+\underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_+^{-T} \hat{\underline N}_i \,,
+			\qquad \underline N_- := -\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.
+		
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="508" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img308.gif"
+ ALT="$\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_...
+...\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(27)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+		Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference element and <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> its affine or bilinear image.
+		Let <!-- MATH
+ $\underline v := \mathcal{P}_K(\hat{\underline v})$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img309.gif"
+ ALT="$ \underline v := \mathcal{P}_K(\hat{\underline v})$"> be a vector field on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the Piola transformation of a reference vector field
+		<!-- MATH
+ $\hat{\underline v} \in \hat{R}$
+ -->
+<IMG
+ WIDTH="50" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img310.gif"
+ ALT="$ \hat{\underline v} \in \hat{R}$">. Let <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> be one of the edges of <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> and <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$">
+		the tangent according to convention <A HREF="node1.html#def:_tangent">1</A>.
+<BR>
+In the case of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> being a triangle, we have <!-- MATH
+ $\hat{R} = \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img91.gif"
+ ALT="$ \hat{R} = \mathcal{R}^k$">. 
+		By the definition of the space <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">, lemma <A HREF="node2.html#lemma:_trace">2</A> and <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> we can
+		conclude that <!-- MATH
+ $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img311.gif"
+ ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
+<BR>
+If <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> is a quadrilateral, the previous remark and 
+		<A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> also tell us that <!-- MATH
+ $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img311.gif"
+ ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
+<BR>
+Hence the condition
+		</FONT><!-- MATH
+ \begin{displaymath}
+\int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underline N_-\cdot \underline t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="401" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img312.gif"
+ ALT="$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underli...
+...ne t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
+$">
+</DIV><P></P><FONT SIZE="-1">
+		on the edge moments
+		is sufficient for the global edge shape functions to satiesfy (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
+		Note that 
+<BR><!-- MATH
+ $\int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$
+ -->
+<IMG
+ WIDTH="307" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img313.gif"
+ ALT="$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$">.
+		So, by the definition (<A HREF="node2.html#eq:_trafo_shape_functions">27</A>) of the element shape functions on <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> resp. on <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$">,
+		by the invariance of the dofs (proposition <A HREF="node2.html#prop:_Dof_invariance">7</A>) and by the definition of the 
+		reference shape functions (example <A HREF="node2.html#ex:_shape_functions">3</A>) we have
+		</FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \int_{\hat{e}_i}  (\hat{\underline N}_i\cdot \hat{\underline t}_i)\hat{\varphi }\,d\hat{s} = 1
+			\qquad \textrm{and} \qquad
+			\int_{e_-} (\underline N_- \cdot \underline t_-)\varphi \,ds = -\int_{\hat{e}_j}  (\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="740" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
+ SRC="img314.gif"
+ ALT="$\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \i...
+...\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	
+</FONT>
+<P>
+<FONT SIZE="-1">To close this section, let us make a note on the interpretation of the dofs on an element <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> in the case of lowest order polynomial
+	degree. In this case, all dofs are edge dofs, the degrees of freedom are <!-- MATH
+ $\hat{\alpha}_j(\hat{\underline
+v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$
+ -->
+<IMG
+ WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img315.gif"
+ ALT="$ \hat{\alpha}_j(\hat{\underline
+v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$"> and the tangential traces of shape functions are constant on each edge.
+	Since we require  <!-- MATH
+ $\hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$
+ -->
+<IMG
+ WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img316.gif"
+ ALT="$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$"> for the reference shape functions, we have 
+	</FONT><!-- MATH
+ \begin{displaymath}
+v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j \cdot \hat{\underline t})\,|\hat{e}_j| = ({\underline N}_j \cdot {\underline t}_j)\, |e_j|\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="313" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img317.gif"
+ ALT="$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j ...
+...\hat{e}_j\vert = ({\underline N}_j \cdot {\underline t}_j)\, \vert e_j\vert\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+	where for the last equality we have used lemma <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A>. We see that
+	the dof <!-- MATH
+ $\alpha_j(\underline v)$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img318.gif"
+ ALT="$ \alpha_j(\underline v)$"> 'sitting' on the edge <IMG
+ WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img319.gif"
+ ALT="$ e_j$"> is the value of the <I>scaled</I> tangential component 
+	<!-- MATH
+ $|e_j|\left(\underline v\cdot \underline t_j\right)|_e$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img320.gif"
+ ALT="$ \vert e_j\vert\left(\underline v\cdot \underline t_j\right)\vert _e$">. 
+<BR>	</FONT><P>
+<DIV><B>R<SMALL>EMARK</SMALL>  6</B> &nbsp; 
+For the invariance of the edge dofs it is essential that the moments <!-- MATH
+ $\alpha^{[K]}$
+ -->
+<IMG
+ WIDTH="38" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img321.gif"
+ ALT="$ \alpha^{[K]}$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> are defined by using the 
+		<I>unit</I> tangent
+		vector <!-- MATH
+ $\underline t = \frac{|\hat{e}|}{|e|}\,(\hat{D}F)\,\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img322.gif"
+ ALT="$ \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert}\,(\hat{D}F)\,\hat{\underline t}$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">. If not, e.&nbsp;g.&nbsp;if we
+		just used the tangent <!-- MATH
+ $\tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="87" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img323.gif"
+ ALT="$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$">, we would lose the invariance of the dofs. In that case the dofs would
+		scale by a factor depending on the size of the edge or face ([<A
+ HREF="node4.html#Ned1">8</A>], remark on p.&nbsp;326).
+<BR>		</DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00025000000000000000">
+2.5 Approximation and convergence results</A>
+</H2><FONT SIZE="-1">
+	Without going into details, we will cite here some results on approximation properties and convergence of N&#233;d&#233;lec FEM of first type.
+	
+<BR>
+We are in the setting of a
+	<I>conforming</I> FEM and have quasi-optimal approximation properties of the FE-spaces <!-- MATH
+ $V_h \subset H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img324.gif"
+ ALT="$ V_h \subset H(\mathop{\rm curl};\Omega )$">
+	</FONT><!-- MATH
+ \begin{displaymath}
+\| \underline u - \Pi_h^k \underline u\|_{H(\mathop{\rm curl}; \Omega )} = C\,\inf_{w\in V_h}\| \underline u -  \underline w\|_{H(\mathop{\rm curl}; \Omega )}\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="355" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img325.gif"
+ ALT="$\displaystyle \Vert \underline u - \Pi_h^k \underline u\Vert _{H(\mathop{\rm cu...
+...V_h}\Vert \underline u - \underline w\Vert _{H(\mathop{\rm curl}; \Omega )}\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+	where <!-- MATH
+ $\Pi_h^k \underline u \in \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="82" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img326.gif"
+ ALT="$ \Pi_h^k \underline u \in \mathcal{R}^k$"> or <!-- MATH
+ $\Pi_h^k \underline u \in \mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img327.gif"
+ ALT="$ \Pi_h^k \underline u \in \mathcal{P}^k$"> respectively,
+	 denotes the interpolate of <!-- MATH
+ $\underline u$
+ -->
+<IMG
+ WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img35.gif"
+ ALT="$ \underline u$"> with regard to the N&#233;d&#233;lec dofs: <!-- MATH
+ $\alpha(\underline u) = \alpha(\Pi_h^k \underline u)$
+ -->
+<IMG
+ WIDTH="121" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img328.gif"
+ ALT="$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$"> for all dofs <IMG
+ WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img329.gif"
+ ALT="$ \alpha$">. 
+	 The interpolation operator <IMG
+ WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img330.gif"
+ ALT="$ \Pi_h^k$"> is defined for sufficiently smooth vector fields, namely for all <!-- MATH
+ $\underline v\in H^r(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img331.gif"
+ ALT="$ \underline v\in H^r(\mathop{\rm curl})$"> 
+	 for any <!-- MATH
+ $r>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img332.gif"
+ ALT="$ r&gt;\frac{1}{2}$"> (see [<A
+ HREF="node4.html#Alonso-Valli">1</A>], Lemma 5.1., [<A
+ HREF="node4.html#Monk">7</A>] and references therein).
+<BR>
+For N&#233;d&#233;lec's FEM of first type we state (without proof) the following optimal estimate in the curl-norm: 
+	</FONT><P>
+<DIV><B>T<SMALL>HEOREM</SMALL>  5</B> &nbsp; 
+If <!-- MATH
+ $\mathcal{T}_h$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img333.gif"
+ ALT="$ \mathcal{T}_h$">, <IMG
+ WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img334.gif"
+ ALT="$ h&gt;0$">, is a regular family of triangulations on <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $"> and <!-- MATH
+ $r>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img332.gif"
+ ALT="$ r&gt;\frac{1}{2}$">, then there exists a constant <IMG
+ WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img335.gif"
+ ALT="$ C&gt;0$">, 
+		depending on <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img336.gif"
+ ALT="$ r$"> but not on <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img337.gif"
+ ALT="$ h$"> or <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">, such that
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl-convergence"></A><!-- MATH
+ \begin{equation}
+\| \underline v - \Pi_h^k \underline v\|_{H(\mathop{\rm curl}; \Omega )} \leq C\,h^{\min\{r,k\}} \|\underline v\|_{H^r(\mathop{\rm curl};\Omega )}\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="348" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img338.gif"
+ ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{H(\mathop{\rm cu...
+... C\,h^{\min\{r,k\}} \Vert\underline v\Vert _{H^r(\mathop{\rm curl};\Omega )}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(28)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for all <!-- MATH
+ $\underline v\in H^r(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img339.gif"
+ ALT="$ \underline v\in H^r(\mathop{\rm curl};\Omega )$">.
+	</DIV><P></P>
+<FONT SIZE="-1">
+	The result in (<A HREF="node2.html#eq:_curl-convergence">28</A>) was obtained by Alonso and Valli in [<A
+ HREF="node4.html#Alonso-Valli">1</A>], extending earlier interpolation  
+	results by N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned1">8</A>] and Monk in [<A
+ HREF="node4.html#Monk_92">6</A>]. 
+<BR>
+Optimal convergence in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-norm for the error of the FE-approximation of the model
+	problem (<A HREF="node1.html#eq:_varform">3</A>) by N&#233;d&#233;lec's elements of first type follows from (<A HREF="node2.html#eq:_curl-convergence">28</A>) by C&#233;a's lemma.
+	This result has been verified in numerical experiments with a <TT>MATLAB</TT> code, which uses lowest
+	order N&#233;d&#233;lec elements on affine triangular meshes for 2d problems, as well as with a  <TT>deal.II</TT> code, which uses lowest order
+	N&#233;d&#233;lec elements on bilinear resp. trilinear meshes for 2d resp. 3d problems.
+<BR>
+As for the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-approximation properties of FE spaces based on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> or <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$">, we could hope for a better order
+	than <!-- MATH
+ $\mathcal{O}(h^k)$
+ -->
+<IMG
+ WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img340.gif"
+ ALT="$ \mathcal{O}(h^k)$"> at first sight: still, we have  <!-- MATH
+ $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$
+ -->
+<IMG
+ WIDTH="165" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img341.gif"
+ ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$">. However, N&#233;d&#233;lec shows in [<A
+ HREF="node4.html#Ned1">8</A>]
+	that only suboptimality can be expected:
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_L2-approx"></A><!-- MATH
+ \begin{equation}
+\| \underline v - \Pi_h^k \underline v\|_{L^2(\Omega )} \leq C h^k |\underline v|_{H^k(\Omega )}\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="244" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img342.gif"
+ ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{L^2(\Omega )} \leq C h^k \vert\underline v\vert _{H^k(\Omega )}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(29)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	N&#233;d&#233;lec uses a standard scaling and Bramble-Hilbert argument to derive (<A HREF="node2.html#eq:_L2-approx">29</A>). Since 
+	<!-- MATH
+ $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$
+ -->
+<IMG
+ WIDTH="254" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img343.gif"
+ ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$">, the Bramble-Hilbert argument only guarantees an elementwise
+	approximation of order <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> of <IMG
+ WIDTH="60" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img344.gif"
+ ALT="$ H^k(K)$">-functions from the space <!-- MATH
+ $\mathcal{R}^k(K)$
+ -->
+<IMG
+ WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img345.gif"
+ ALT="$ \mathcal{R}^k(K)$">.
+<BR>
+However, in a recent paper Hiptmair uses a duality technique to state optimal convergence of the
+	<!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-error <!-- MATH
+ $\| \underline u - \underline u_h\|_{L^2(\Omega )}$
+ -->
+<IMG
+ WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img346.gif"
+ ALT="$ \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )}$">
+	for the 3d case and N&#233;d&#233;lec's elements of first type of order <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> on tetrahedral meshes 
+	(see Section 5.3, Theorem 5.8 in [<A
+ HREF="node4.html#Hipt">5</A>]): 
+	</FONT><P>
+<DIV><A NAME="th:_Hiptmair"><B>T<SMALL>HEOREM</SMALL>  6</B></A> &nbsp; 
+There is an <!-- MATH
+ $s>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img347.gif"
+ ALT="$ s&gt;\frac{1}{2}$"> such that
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_L2-convergence"></A><!-- MATH
+ \begin{equation}
+\| \underline u -  \underline u_h\|_{L^2(\Omega )} \leq C h^s \| \underline u - \underline u_h\|_{H(\mathop{\rm curl}; \Omega )}\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="299" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img348.gif"
+ ALT="$\displaystyle \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )} \leq C h^s \Vert \underline u - \underline u_h\Vert _{H(\mathop{\rm curl}; \Omega )}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(30)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+Under the assumption that the boundary <!-- MATH
+ $\partial \Omega$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img9.gif"
+ ALT="$ \partial \Omega $"> is smooth or convex, <IMG
+ WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img349.gif"
+ ALT="$ s=1$"> can be chosen.
+	</DIV><P></P>
+<FONT SIZE="-1">
+	Several key arguments of the proof in [<A
+ HREF="node4.html#Hipt">5</A>] make explicitely use of features that are limited to 3d problems and the 
+	family of finite elements based on tetrahedrons.
+	They cannot be modified trivially to apply to 2d problems or 3d problems on hexahedral meshes. Even worse, it is suggested by the results of
+	numerical experiments that one cannot hope to obtain a result similar to (<A HREF="node2.html#eq:_L2-convergence">30</A>).
+<BR>
+A possibility to overcome this deficiency of convergence is to use N&#233;d&#233;lec elements of second type, where the full <!-- MATH
+ $[\mathbb{P}_k]^d$
+ -->
+<IMG
+ WIDTH="43" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img350.gif"
+ ALT="$ [\mathbb{P}_k]^d$">
+	are used as polynomial spaces (see [<A
+ HREF="node4.html#Ned2">10</A>]).
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
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+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>3 Numerical results</TITLE>
+<META NAME="description" CONTENT="3 Numerical results">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
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+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+
+<H1><A NAME="SECTION00030000000000000000">
+3 <FONT SIZE="+1">Numerical results</FONT></A>
+</H1><FONT SIZE="-1">
+	The numerical results in this section provide some samples of the quality of the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM 
+	with N&#233;d&#233;lec elements of first type and lowest order (polynomial degree <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">).
+<BR>
+We considered the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in <!-- MATH
+ $\Omega = [-1,1]^d$
+ -->
+<IMG
+ WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img351.gif"
+ ALT="$ \Omega = [-1,1]^d$">, <IMG
+ WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ d=2,3$">, 
+	with homogeneous Dirichlet boundary condition (<A HREF="node1.html#eq:_PCB">2</A>).
+<BR>
+The first few results for the two-dimensional problem have been obtained by a <TT>MATLAB</TT> code. 
+	For the first example we used the data 
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_numerical_ex.1"></A><!-- MATH
+ \begin{equation}
+c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\3 - x^2 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="265" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img352.gif"
+ ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\  3 - x^2 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(31)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	For the second example we have followed the outlines from Appendix A and taken the data from example <A HREF="node5.html#ex:_solutions_from_Laplace">5</A>
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_numerical_ex.2"></A><!-- MATH
+ \begin{equation}
+c\equiv 1 \,, \qquad
+		\underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,.
+	
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="401" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img353.gif"
+ ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\  -\sin\pi x\cos\pi y \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(32)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+	The finite element solution has been computed using N&#233;d&#233;lec elements of first type and of polynomial degree <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$"> on a family of
+	affine triangular grids.
+	The initial coarse grid consisted of <IMG
+ WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img354.gif"
+ ALT="$ 2^5$"> triangles. The finest grid with <IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img355.gif"
+ ALT="$ 2^{13}$"> triangles results after five global refinements.
+<BR>
+In Table <A HREF="#table:_matlab">1</A> we see that for both examples we have 
+	optimal convergence in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-semiorm, as we would expect from the theoretical results of the
+	previous section. As for the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm, it appears that in both examples the convergence
+	of the numerical solution is not optimal for our choice of finite elements. In the case of N&#233;d&#233;lec elements of first 
+	type and of polynomial degree <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">, we got only <!-- MATH
+ $\mathcal{O}(h)$
+ -->
+<IMG
+ WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img356.gif"
+ ALT="$ \mathcal{O}(h)$">-convergence of the <IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img357.gif"
+ ALT="$ L^2$">-error. However, this order of
+	convergence is consistent with the result (<A HREF="node2.html#eq:_L2-approx">29</A>) obtained by N&#233;d&#233;lec in [<A
+ HREF="node4.html#Ned1">8</A>].
+<BR></FONT>
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER"><A NAME="4142"></A>
+<TABLE>
+<CAPTION><STRONG>Table 1:</STRONG>
+Errors and convergence rates in the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm and <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img359.gif"
+ ALT="$ H(\mathop{\rm curl};\Omega )$">-seminorm for the two <TT>MATLAB</TT> examples.</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">grid</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img358.gif"
+ ALT="$ \char93 $"> cells</TD>
+<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
+ $H(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img46.gif"
+ ALT="$ H(\mathop{\rm curl})$">-error</TD>
+<TD ALIGN="CENTER" COLSPAN=2><IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img357.gif"
+ ALT="$ L^2$">-error</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">1</TD>
+<TD ALIGN="RIGHT">32</TD>
+<TD ALIGN="CENTER">6.66e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+<TD ALIGN="CENTER">4.66e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">2</TD>
+<TD ALIGN="RIGHT">128</TD>
+<TD ALIGN="CENTER">3.33e-01</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">2.35e-01</TD>
+<TD ALIGN="CENTER">0.98</TD>
+</TR>
+<TR><TD ALIGN="CENTER">example 1</TD>
+<TD ALIGN="CENTER">3</TD>
+<TD ALIGN="RIGHT">512</TD>
+<TD ALIGN="CENTER">1.66e-01</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">1.17e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">4</TD>
+<TD ALIGN="RIGHT">2048</TD>
+<TD ALIGN="CENTER">8.33e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">5.89e-02</TD>
+<TD ALIGN="CENTER">0.99</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">5</TD>
+<TD ALIGN="RIGHT">8192</TD>
+<TD ALIGN="CENTER">4.17e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">2.95e-02</TD>
+<TD ALIGN="CENTER">0.99</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">1</TD>
+<TD ALIGN="RIGHT">32</TD>
+<TD ALIGN="CENTER">3.05e+00</TD>
+<TD ALIGN="CENTER">-</TD>
+<TD ALIGN="CENTER">6.48e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">2</TD>
+<TD ALIGN="RIGHT">128</TD>
+<TD ALIGN="CENTER">1.61e+00</TD>
+<TD ALIGN="CENTER">0.91</TD>
+<TD ALIGN="CENTER">3.22e-01</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+<TR><TD ALIGN="CENTER">example 2</TD>
+<TD ALIGN="CENTER">3</TD>
+<TD ALIGN="RIGHT">512</TD>
+<TD ALIGN="CENTER">0.81e-01</TD>
+<TD ALIGN="CENTER">0.97</TD>
+<TD ALIGN="CENTER">1.60e-01</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">4</TD>
+<TD ALIGN="RIGHT">2048</TD>
+<TD ALIGN="CENTER">0.41e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+<TD ALIGN="CENTER">8.02e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">5</TD>
+<TD ALIGN="RIGHT">8192</TD>
+<TD ALIGN="CENTER">2.05e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+<TD ALIGN="CENTER">4.01e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+</TABLE>
+</DIV>
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+<BR>
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="fig:conv1"></A><A NAME="4143"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
+Convergence of the FE-approximation to the smooth solution of the <TT>MATLAB</TT> example (<A HREF="node3.html#eq:_numerical_ex.1">31</A>) 
+	in the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm and the <!-- MATH
+ $H(\mathop{\rm curl};(\Omega ))$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img361.gif"
+ ALT="$ H(\mathop{\rm curl};(\Omega ))$">-seminorm</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}$
+ -->
+<IMG
+ WIDTH="431" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
+ SRC="img360.gif"
+ ALT="% latex2html id marker 4265
+\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}">
+    
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="fig:conv2"></A><A NAME="4144"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
+Convergence of the FE-approximation to the smooth solution of the <TT>MATLAB</TT> example (<A HREF="node3.html#eq:_numerical_ex.2">32</A>) 
+	in the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm and the <!-- MATH
+ $H(\mathop{\rm curl};(\Omega ))$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img361.gif"
+ ALT="$ H(\mathop{\rm curl};(\Omega ))$">-seminorm</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}$
+ -->
+<IMG
+ WIDTH="431" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
+ SRC="img362.gif"
+ ALT="% latex2html id marker 4271
+\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}">
+    
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL>  7</B> &nbsp; 
+The mesh generation and refinement was done by <TT>PDE-toolbox</TT> commands. Since the <TT>PDE-toolbox</TT> does not support three
+		dimensional grids, we restricted ourselves to 2d problems, and we have so far no numerical results for the case of
+		tetrahedral grids in 3d.
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">As for meshes with quadrilateral cells, numerical results were obtained with a <TT>deal.II</TT> code,  
+	using the finite element class <TT>fe/fenedelec.cc</TT>. This class provides 
+	N&#233;d&#233;lec's <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming element of first type and lowest order in two and three space dimensions, on 
+	bilinear quadrilateral, resp. trilinear hexahedral grids. For details about <TT>deal.II</TT>, see [<A
+ HREF="node4.html#Deal">2</A>].
+	In the following results were obtained for the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in two dimensions using the 
+	data (<A HREF="node3.html#eq:_numerical_ex.2">32</A>).  
+	We computed the solution on five successive
+	non-affine bilinear grids ( figure <A HREF="node3.html#fig:grid">3</A>), each of which was obtained by global refinement of the previous one.
+</FONT>
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="fig:grid"></A><A NAME="4146"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
+Non-affine bilinear grid used in the <TT>deal.II</TT> code, after one refinement step.</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}$
+ -->
+<IMG
+ WIDTH="249" HEIGHT="249" ALIGN="BOTTOM" BORDER="0"
+ SRC="img363.gif"
+ ALT="\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}">
+    
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Again, in Table <A HREF="#table:_2d-deal">2</A> we can observe optimal convergence of order <!-- MATH
+ $\mathcal{O}(h)$
+ -->
+<IMG
+ WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img356.gif"
+ ALT="$ \mathcal{O}(h)$"> in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-norm. 
+The same order of convergence is obtained for the error in the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm.
+<BR></FONT>
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER"><A NAME="4147"></A>
+<TABLE>
+<CAPTION><STRONG>Table 2:</STRONG>
+Errors and convergence rates in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img359.gif"
+ ALT="$ H(\mathop{\rm curl};\Omega )$">- and <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm for the 2d-example solved with <TT>deal.II</TT>.
+</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
+<TR><TD ALIGN="CENTER">grid</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img358.gif"
+ ALT="$ \char93 $"> cells</TD>
+<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
+ $H(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img46.gif"
+ ALT="$ H(\mathop{\rm curl})$">-error</TD>
+<TD ALIGN="CENTER" COLSPAN=2><IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img357.gif"
+ ALT="$ L^2$">-error</TD>
+</TR>
+<TR><TD ALIGN="CENTER">1</TD>
+<TD ALIGN="RIGHT">4</TD>
+<TD ALIGN="CENTER">6.112e+00</TD>
+<TD ALIGN="CENTER">-</TD>
+<TD ALIGN="CENTER">1.442e+00</TD>
+<TD ALIGN="CENTER">-</TD>
+</TR>
+<TR><TD ALIGN="CENTER">2</TD>
+<TD ALIGN="RIGHT">16</TD>
+<TD ALIGN="CENTER">3.688e+00</TD>
+<TD ALIGN="CENTER">0.73</TD>
+<TD ALIGN="CENTER">6.765e-01</TD>
+<TD ALIGN="CENTER">1.09</TD>
+</TR>
+<TR><TD ALIGN="CENTER">3</TD>
+<TD ALIGN="RIGHT">64</TD>
+<TD ALIGN="CENTER">1.991e+00</TD>
+<TD ALIGN="CENTER">0.89</TD>
+<TD ALIGN="CENTER">3.280e-01</TD>
+<TD ALIGN="CENTER">1.04</TD>
+</TR>
+<TR><TD ALIGN="CENTER">4</TD>
+<TD ALIGN="RIGHT">256</TD>
+<TD ALIGN="CENTER">1.015e+00</TD>
+<TD ALIGN="CENTER">0.97</TD>
+<TD ALIGN="CENTER">1.617e-01</TD>
+<TD ALIGN="CENTER">1.02</TD>
+</TR>
+<TR><TD ALIGN="CENTER">5</TD>
+<TD ALIGN="RIGHT">1024</TD>
+<TD ALIGN="CENTER">5.098e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+<TD ALIGN="CENTER">8.049e-02</TD>
+<TD ALIGN="CENTER">1.01</TD>
+</TR>
+</TABLE>
+</DIV>
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+<BR>
+<P>
+<FONT SIZE="-1">With <TT>deal.II</TT>, we are also able to treat 3d problems on hexahedral grids. For our type of problem, 
+N&#233;d&#233;lec's <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming elements of first type and lowest order, based on a cubic reference element, are available. 
+<BR>
+We computed an approximation to the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in 3d using the data
+	</FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{ccc}   xy(1 - y^2)(1-z^2) + 2xy(1-z^2) \\
+																	  y^2(1 - x^2)(1-z^2) + (1-y^2)(2-x^2-z^2) \\
+																	  yz(1 - x^2)(1-y^2) + 2yz(1-x^2)
+											 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="557" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img364.gif"
+ ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{cc...
+... + (1-y^2)(2-x^2-z^2) \\  yz(1 - x^2)(1-y^2) + 2yz(1-x^2) \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(33)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+In a first experiment, the finite element solution was computed on five successive globally refined affine grids. In a second 
+computation, we approximated the solution of the same problem on five successive globally refined non-affine trilinear grids.
+<BR>
+We see in Table <A HREF="#table:_3d-deal">3</A> that in both cases we observe again convergence of order <!-- MATH
+ $\mathcal{O}(h)$
+ -->
+<IMG
+ WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img356.gif"
+ ALT="$ \mathcal{O}(h)$"> in the  
+<!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">- and the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm.
+<BR></FONT>
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER"><A NAME="4148"></A>
+<TABLE>
+<CAPTION><STRONG>Table 3:</STRONG>
+Errors and convergence rates in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img359.gif"
+ ALT="$ H(\mathop{\rm curl};\Omega )$">- and <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-norm for the 3d-example solved with <TT>deal.II</TT>. The first data
+set is for the computation on a family of affine grids, the second set of data is for non-affine trilinear grids.</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">grid</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img358.gif"
+ ALT="$ \char93 $"> cells</TD>
+<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
+ $H(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img46.gif"
+ ALT="$ H(\mathop{\rm curl})$">-error</TD>
+<TD ALIGN="CENTER" COLSPAN=2><IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img357.gif"
+ ALT="$ L^2$">-error</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">1</TD>
+<TD ALIGN="RIGHT">8</TD>
+<TD ALIGN="CENTER">7.696e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+<TD ALIGN="CENTER">6.609e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">2</TD>
+<TD ALIGN="RIGHT">64</TD>
+<TD ALIGN="CENTER">4.088e-01</TD>
+<TD ALIGN="CENTER">0.91</TD>
+<TD ALIGN="CENTER">2.943e-01</TD>
+<TD ALIGN="CENTER">1.17</TD>
+</TR>
+<TR><TD ALIGN="CENTER">affine grids</TD>
+<TD ALIGN="CENTER">3</TD>
+<TD ALIGN="RIGHT">512</TD>
+<TD ALIGN="CENTER">2.075e-01</TD>
+<TD ALIGN="CENTER">0.98</TD>
+<TD ALIGN="CENTER">1.408e-01</TD>
+<TD ALIGN="CENTER">1.06</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">4</TD>
+<TD ALIGN="RIGHT">4096</TD>
+<TD ALIGN="CENTER">1.041e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+<TD ALIGN="CENTER">6.955e-02</TD>
+<TD ALIGN="CENTER">1.02</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">5</TD>
+<TD ALIGN="RIGHT">32768</TD>
+<TD ALIGN="CENTER">5.210e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">3.467e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">1</TD>
+<TD ALIGN="RIGHT">8</TD>
+<TD ALIGN="CENTER">7.716e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+<TD ALIGN="CENTER">6.611e-01</TD>
+<TD ALIGN="CENTER">-</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">2</TD>
+<TD ALIGN="RIGHT">64</TD>
+<TD ALIGN="CENTER">4.108e-01</TD>
+<TD ALIGN="CENTER">0.91</TD>
+<TD ALIGN="CENTER">2.955e-01</TD>
+<TD ALIGN="CENTER">1.16</TD>
+</TR>
+<TR><TD ALIGN="CENTER">non-affine grids</TD>
+<TD ALIGN="CENTER">3</TD>
+<TD ALIGN="RIGHT">512</TD>
+<TD ALIGN="CENTER">2.085e-01</TD>
+<TD ALIGN="CENTER">0.98</TD>
+<TD ALIGN="CENTER">1.413e-01</TD>
+<TD ALIGN="CENTER">1.06</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">4</TD>
+<TD ALIGN="RIGHT">4096</TD>
+<TD ALIGN="CENTER">1.046e-01</TD>
+<TD ALIGN="CENTER">0.99</TD>
+<TD ALIGN="CENTER">6.982e-02</TD>
+<TD ALIGN="CENTER">1.02</TD>
+</TR>
+<TR><TD ALIGN="CENTER">&nbsp;</TD>
+<TD ALIGN="CENTER">5</TD>
+<TD ALIGN="RIGHT">32768</TD>
+<TD ALIGN="CENTER">5.237e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+<TD ALIGN="CENTER">3.480e-02</TD>
+<TD ALIGN="CENTER">1.00</TD>
+</TR>
+</TABLE>
+</DIV>
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+<BR><FONT SIZE="-1">
+The conclusion that can be drawn from these numerical experiments is, that the restriction to three-dimensional tetrahedral grids 
+of Hiptmair's result on the <IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img357.gif"
+ ALT="$ L^2$">-convergence of the error (<A HREF="node2.html#th:_Hiptmair">6</A>) cannot be relaxed. 
+<BR></FONT>
+<P>
+<FONT SIZE="-1">Finally, here are some pretty pictures: the vector field plots from the <TT>MATLAB</TT> computations. 
+</FONT>
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="fig:field1"></A><A NAME="4149"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
+Vector-field plot of the FE-solution of example (<A HREF="node3.html#eq:_numerical_ex.1">31</A>).</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=9.5cm, height=7cm]{field1.eps}$
+ -->
+<IMG
+ WIDTH="432" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
+ SRC="img365.gif"
+ ALT="\includegraphics[width=9.5cm, height=7cm]{field1.eps}">
+    
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="fig:field2"></A><A NAME="4150"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
+Vector-field plot of the FE-solution of example (<A HREF="node3.html#eq:_numerical_ex.2">32</A>).</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=9.5cm, height=7cm]{field2.eps}$
+ -->
+<IMG
+ WIDTH="432" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
+ SRC="img366.gif"
+ ALT="\includegraphics[width=9.5cm, height=7cm]{field2.eps}">
+    
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+<FONT SIZE="-1">
+</FONT>
+<P>
+
+<P>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
diff --git a/deal.II/doc/reports/nedelec/node4.html b/deal.II/doc/reports/nedelec/node4.html
new file mode 100644
index 0000000000..5c75f9284a
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/node4.html
@@ -0,0 +1,139 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>Bibliography</TITLE>
+<META NAME="description" CONTENT="Bibliography">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+
+<H2><A NAME="SECTION00040000000000000000">
+Bibliography</A>
+</H2><DL COMPACT><DD><P></P><DT><A NAME="Alonso-Valli">1</A>
+<DD>
+A.&nbsp;Alonso and A.&nbsp;Valli.
+<BR>An optimal domain decomposition preconditioner for low-frequency
+  time-harmonic Maxwell equations.
+<BR><EM>Math.&nbsp;Comp.</EM>, 68(226):607-631, 1999.
+
+<P></P><DT><A NAME="Deal">2</A>
+<DD>
+W.&nbsp;Bangerth, R.&nbsp;Hartmann, and G.&nbsp;Kanschat.
+<BR><EM><TT>deal.II</TT> Differential Equations Analysis Library,
+  Technical Reference</EM>.
+<BR>IWR, Universit&#228;t Heidelberg.
+<BR><TT>http://www.dealii.org</TT>.
+
+<P></P><DT><A NAME="Brezzi-Fortin">3</A>
+<DD>
+F.&nbsp;Brezzi and M.&nbsp;Fortin.
+<BR><EM>Mixed and Hybrid Finite Element Methods</EM>, volume&nbsp;15 of <EM>  Springer Series in Computational Mathematics</EM>.
+<BR>Springer-Verlag, New York, 1991.
+
+<P></P><DT><A NAME="Girault-Raviart">4</A>
+<DD>
+V.&nbsp;Girault and P.-A. Raviart.
+<BR><EM>Finite Element Approximation of the Navier-Stokes Equations</EM>,
+  volume 749 of <EM>Lecture Notes in Mathematics</EM>.
+<BR>Springer-Verlag, Berlin, Heidelberg, 1979, 1981.
+
+<P></P><DT><A NAME="Hipt">5</A>
+<DD>
+R.&nbsp;Hiptmair.
+<BR>Finite elements in computational electromagnetism.
+<BR>In <EM>Acta Numerica</EM>, pages 1-103. Cambridge University press,
+  2002.
+
+<P></P><DT><A NAME="Monk_92">6</A>
+<DD>
+P.&nbsp;Monk.
+<BR>Analysis of a finite element method for Maxwell's equations.
+<BR><EM>SIAM J.&nbsp;Numer.&nbsp;Anal</EM>, 29:714-729, 1992.
+
+<P></P><DT><A NAME="Monk">7</A>
+<DD>
+P.&nbsp;Monk.
+<BR>A simple proof for an edge element discretization of Maxwell's
+  equations.
+<BR>Submitted for publication. Download version available on Monk's
+  webpage: www.math.udel.edu./&nbsp;monk, 2001.
+
+<P></P><DT><A NAME="Ned1">8</A>
+<DD>
+J.&nbsp;C. N&#233;d&#233;lec.
+<BR>Mixed finite elements in <!-- MATH
+ $\mathbb{R}^3$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img367.gif"
+ ALT="$ \mathbb{R}^3$">.
+<BR><EM>Numer.&nbsp;Math.</EM>, 35:315-341, 1980.
+
+<P></P><DT><A NAME="Ned3">9</A>
+<DD>
+J.&nbsp;C. N&#233;d&#233;lec.
+<BR>Elements finis mixtes incompressibles pour l'&#233;quation de Stokes
+  dans <!-- MATH
+ $\mathbb{R}^3$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img367.gif"
+ ALT="$ \mathbb{R}^3$">.
+<BR><EM>Numer.&nbsp;Math.</EM>, 39:97-112, 1982.
+
+<P></P><DT><A NAME="Ned2">10</A>
+<DD>
+J.&nbsp;C. N&#233;d&#233;lec.
+<BR>A new family of mixed finite elements in <!-- MATH
+ $\mathbb{R}^3$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img367.gif"
+ ALT="$ \mathbb{R}^3$">.
+<BR><EM>Numer.&nbsp;Math.</EM>, 50:57-81, 1986.
+
+<P></P><DT><A NAME="Demko">11</A>
+<DD>
+W.&nbsp;Rachowicz and L.&nbsp;Demkowicz.
+<BR>A two-dimensional hp-adaptive finite element package for
+  electromagnetics (2Dhp90_EM).
+<BR>Ticam Report 98-16, TICAM, 1998.
+<BR>Download version available on Demkowicz' webpage:
+  www.ticam.utexas.edu/&nbsp;Leszek.
+
+<P></P><DT><A NAME="Demko3d">12</A>
+<DD>
+W.&nbsp;Rachowicz and L.&nbsp;Demkowicz.
+<BR>A three-dimensional hp-adaptive finite element package for
+  electromagnetics (3Dhp90_EM).
+<BR>Ticam Report 00-04.2000, TICAM, 2000.
+<BR>Download version available on Demkowicz' webpage:
+  www.ticam.utexas.edu/&nbsp;Leszek.
+</DL>
+
+</FONT><BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
diff --git a/deal.II/doc/reports/nedelec/node5.html b/deal.II/doc/reports/nedelec/node5.html
new file mode 100644
index 0000000000..7568b72569
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/node5.html
@@ -0,0 +1,332 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>A. Construction of solutions in 2d</TITLE>
+<META NAME="description" CONTENT="A. Construction of solutions in 2d">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+
+<H1><A NAME="SECTION00050000000000000000">
+A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
+</H1>
+<P>
+<FONT SIZE="-1">We present how divergence-free solutions of the model problem (<A HREF="node1.html#eq:model_problem">1</A>) on a domain <!-- MATH
+ $\Omega \subset\mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img167.gif"
+ ALT="$ \Omega \subset\mathbb{R}^2$"> with perfectly
+	conducting boundary
+	can be constructed from solutions of the scalar Laplace equation.
+<BR></FONT>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL>  9</B> &nbsp; 
+Let <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $"> be a sufficiently smooth domain in <!-- MATH
+ $\mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img368.gif"
+ ALT="$ \mathbb{R}^2$">, <!-- MATH
+ $\varphi (x,y)$
+ -->
+<IMG
+ WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img369.gif"
+ ALT="$ \varphi (x,y)$"> a sufficently smooth scalar function on <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $"> and the coefficient <IMG
+ WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img370.gif"
+ ALT="$ c&gt;0$">
+		globally constant.
+<BR>
+Let <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img371.gif"
+ ALT="$ w$"> be a solution of the scalar equation
+		<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Laplace"></A><!-- MATH
+ \begin{equation}
+\begin{split}
+			-\Delta w + c\, w &= \varphi \quad \mathrm{in} \quad \Omega \\
+			\underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,.
+		\end{split}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="382" HEIGHT="55" BORDER="0"
+ SRC="img372.gif"
+ ALT="\begin{displaymath}\begin{split}-\Delta w + c\, w &amp;= \varphi \quad \mathrm{in} \...
+... w &amp;= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(34)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+Then, <!-- MATH
+ $\underline E := \nabla^{\perp} w$
+ -->
+<IMG
+ WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img373.gif"
+ ALT="$ \underline E := \nabla^{\perp} w$"> is a solution of the model equation
+		<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+			\underline \mathop{\rm curl}\mathop{\rm curl}\underline E + c\, \underline E  = \underline f \quad \mathrm{in} \quad \Omega \,, \\
+			\underline E \wedge \underline n = 0 \quad \mathrm{on} \quad \partial \Omega \,,
+		\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="397" HEIGHT="60" BORDER="0"
+ SRC="img374.gif"
+ ALT="\begin{displaymath}\begin{split}\underline \mathop{\rm curl}\mathop{\rm curl}\un...
+...e n = 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+with right hand side <!-- MATH
+ $\underline f := \nabla^{\perp} \varphi$
+ -->
+<IMG
+ WIDTH="82" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img375.gif"
+ ALT="$ \underline f := \nabla^{\perp} \varphi $">. 
+<BR>
+We use the notation <!-- MATH
+ $\nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\-\partial _x\varphi
+\end{array}\right)$
+ -->
+<IMG
+ WIDTH="220" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img376.gif"
+ ALT="$ \nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\  -\partial _x\varphi
+\end{array}\right)$">.
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+		We first show the correspondence of the boundary conditions. With the definition <!-- MATH
+ $\underline E := \nabla^{\perp} w$
+ -->
+<IMG
+ WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img373.gif"
+ ALT="$ \underline E := \nabla^{\perp} w$"> it holds
+		</FONT><!-- MATH
+ \begin{displaymath}
+\underline E \wedge \underline n = \underline E \cdot \underline t = {\nabla w}^T \boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="303" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img377.gif"
+ ALT="$\displaystyle \underline E \wedge \underline n = \underline E \cdot \underline ...
+...boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+		It remains to show that <!-- MATH
+ $\underline E$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img378.gif"
+ ALT="$ \underline E$"> solves the model problem for an appropriate right hand side. 
+		First, note that <!-- MATH
+ $\underline E$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img378.gif"
+ ALT="$ \underline E$"> is divergence-free:
+		<!-- MATH
+ $\nabla\cdot\nabla^{\perp}w = 0$
+ -->
+<IMG
+ WIDTH="104" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img379.gif"
+ ALT="$ \nabla\cdot\nabla^{\perp}w = 0$"> for all <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img371.gif"
+ ALT="$ w$">. Hence, the identity <!-- MATH
+ $\underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$
+ -->
+<IMG
+ WIDTH="229" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img380.gif"
+ ALT="$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$">
+		reduces to <!-- MATH
+ $\underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$
+ -->
+<IMG
+ WIDTH="152" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img381.gif"
+ ALT="$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$">. The observation that for smooth data <!-- MATH
+ $\nabla^{\perp}w$
+ -->
+<IMG
+ WIDTH="45" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img382.gif"
+ ALT="$ \nabla^{\perp}w$"> solves the Laplace equation
+		(<A HREF="node5.html#eq:_Laplace">34</A>) with right hand side <!-- MATH
+ $\nabla^{\perp} \varphi$
+ -->
+<IMG
+ WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img383.gif"
+ ALT="$ \nabla^{\perp} \varphi $"> concludes the proof.
+	
+</FONT>
+<P>
+<P>
+<DIV><A NAME="ex:_solutions_from_Laplace"><B>E<SMALL>XAMPLE</SMALL>  5</B></A> (Solutions from eigenfunctions of the Laplacian)  &nbsp; 
+Choose <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img371.gif"
+ ALT="$ w$"> to be a solution of the eigenvalue problem
+		<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+			-\Delta w  &= \lambda \, w \quad \mathrm{in} \quad \Omega \\
+			\underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,,
+		\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="364" HEIGHT="55" BORDER="0"
+ SRC="img384.gif"
+ ALT="\begin{displaymath}\begin{split}-\Delta w &amp;= \lambda \, w \quad \mathrm{in} \qua...
+... w &amp;= 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+and set <!-- MATH
+ $\varphi = (\lambda + c)\,w$
+ -->
+<IMG
+ WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img385.gif"
+ ALT="$ \varphi = (\lambda + c)\,w$">.
+<BR>
+As an example, take <!-- MATH
+ $\Omega = [-1,1]^2$
+ -->
+<IMG
+ WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img386.gif"
+ ALT="$ \Omega = [-1,1]^2$"> and <!-- MATH
+ $\lambda = 2\pi^2$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img387.gif"
+ ALT="$ \lambda = 2\pi^2$">. Then, <!-- MATH
+ $w = \cos\pi x\cos\pi y$
+ -->
+<IMG
+ WIDTH="138" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img388.gif"
+ ALT="$ w = \cos\pi x\cos\pi y$"> is an eigenfunction and we compute 
+		<!-- MATH
+ \begin{displaymath}
+\underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,, \qquad
+			\underline E = \pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="536" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img389.gif"
+ ALT="$\displaystyle \underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\...
+...in{array}{cc} \cos\pi x\sin\pi y \\  -\sin\pi x\cos\pi y \end{array}\right)\,.
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<P>
+<DIV><B>E<SMALL>XAMPLE</SMALL>  6</B> (Solutions from any scalar function satiesfying the boundary condition)  &nbsp; 
+Take again <!-- MATH
+ $\Omega = [-1,1]^2$
+ -->
+<IMG
+ WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img386.gif"
+ ALT="$ \Omega = [-1,1]^2$">. We have to find a scalar function <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img371.gif"
+ ALT="$ w$"> which satiesfies the homogeneous Neumann boundary condition. Take
+		for example <!-- MATH
+ $w(x,y) = (1-x^2)^2(1-y^2)^2$
+ -->
+<IMG
+ WIDTH="220" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img390.gif"
+ ALT="$ w(x,y) = (1-x^2)^2(1-y^2)^2$">, for which we have <!-- MATH
+ $\underline n \cdot \nabla w = 0$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img391.gif"
+ ALT="$ \underline n \cdot \nabla w = 0$"> on <!-- MATH
+ $\partial [-1,1]^2$
+ -->
+<IMG
+ WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img392.gif"
+ ALT="$ \partial [-1,1]^2$">. The right hand side is then 
+	    <!-- MATH
+ $\varphi = -\Delta w + c w$
+ -->
+<IMG
+ WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img393.gif"
+ ALT="$ \varphi = -\Delta w + c w$">. 
+	</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">
+</FONT>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
diff --git a/deal.II/doc/reports/nedelec/node6.html b/deal.II/doc/reports/nedelec/node6.html
new file mode 100644
index 0000000000..423df4c2fa
--- /dev/null
+++ b/deal.II/doc/reports/nedelec/node6.html
@@ -0,0 +1,310 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by:  Nikos Drakos, CBLU, University of Leeds
+* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>A. Time-harmonic Maxwell's equations with low-frequency approximation</TITLE>
+<META NAME="description" CONTENT="A. Time-harmonic Maxwell's equations with low-frequency approximation">
+<META NAME="keywords" CONTENT="main">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
+
+<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta">
+<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
+
+<LINK REL="STYLESHEET" HREF="main.css">
+
+</HEAD>
+
+<BODY >
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
+
+<UL>
+<LI><A NAME="tex2html110"
+  HREF="#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
+</UL>
+<!--End of Table of Child-Links-->
+<HR>
+
+<H1><A NAME="SECTION00060000000000000000"></A> 
+<A NAME="appendix:_Maxwell_s_Eq."></A>
+<BR>
+A. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT>
+</H1><FONT SIZE="-1">
+We show, how the model problem can be derived from the time-harmonic Maxwell's equations in the low-frequency case.
+We follow the outline of [<A
+ HREF="node4.html#Alonso-Valli">1</A>]:
+<BR>
+We consider the following primal formulation of Maxwell's equations: 
+</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_maxwell"></A><!-- MATH
+ \begin{equation}
+\begin{split}
+\varepsilon \frac{\partial \mathcal{E}}{\partial t} & = \mathop{\rm curl}\mathcal{H} - \sigma \mathcal{E} \,, \\
+	\mu \frac{\partial \mathcal{H}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,,
+\end{split}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="361" HEIGHT="85" BORDER="0"
+ SRC="img394.gif"
+ ALT="\begin{displaymath}\begin{split}\varepsilon \frac{\partial \mathcal{E}}{\partial...
+...}}{\partial t} &amp; = -\mathop{\rm curl}\mathcal{E}\,, \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(35)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+where <!-- MATH
+ $\mathcal{E}$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img395.gif"
+ ALT="$ \mathcal{E}$"> and <!-- MATH
+ $\mathcal{H}$
+ -->
+<IMG
+ WIDTH="21" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img396.gif"
+ ALT="$ \mathcal{H}$"> are the electric and magnetic field. 
+<!-- MATH
+ $\varepsilon (x), \mu(x)$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img397.gif"
+ ALT="$ \varepsilon (x), \mu(x)$"> are the dielectric and magnetic permeability coefficients, and <IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img398.gif"
+ ALT="$ \sigma(x)$"> denotes the electric conductivity.
+<!-- MATH
+ $\varepsilon (x), \mu(x)$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img397.gif"
+ ALT="$ \varepsilon (x), \mu(x)$"> and <IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img398.gif"
+ ALT="$ \sigma(x)$"> are assumed to be symmetric matrices in <!-- MATH
+ $L^{\infty}(\Omega )^{d\times d}$
+ -->
+<IMG
+ WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img399.gif"
+ ALT="$ L^{\infty}(\Omega )^{d\times d}$">, and <!-- MATH
+ $\varepsilon (x)$
+ -->
+<IMG
+ WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img400.gif"
+ ALT="$ \varepsilon (x)$"> and <IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img401.gif"
+ ALT="$ \mu(x)$"> are
+positive definite. <IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img398.gif"
+ ALT="$ \sigma(x)$"> is positive definite in a conductor and vanishes in an insulator.
+</FONT>
+<P>
+
+<H2><A NAME="SECTION00061000000000000000">
+Time-harmonic, low-frequency case</A>
+</H2><FONT SIZE="-1">
+We assume that <!-- MATH
+ $\mathcal{E}(x,t)$
+ -->
+<IMG
+ WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img402.gif"
+ ALT="$ \mathcal{E}(x,t)$"> and <!-- MATH
+ $\mathcal{H}(x,t)$
+ -->
+<IMG
+ WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img403.gif"
+ ALT="$ \mathcal{H}(x,t)$"> are <I>time-harmonic</I>, i.&nbsp;e.&nbsp;they can be represented as
+</FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+	\mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(i\omega t)\right) \,, \\
+	\mathcal{H}(x,t) &= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,.
+\end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="396" HEIGHT="55" BORDER="0"
+ SRC="img404.gif"
+ ALT="\begin{displaymath}\begin{split}\mathcal{E}(x,t) &amp;= \mathrm{Re} \left(E(x) \exp(...
+...= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+&nbsp;&nbsp;&nbsp;</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+Here, <!-- MATH
+ $E(x), H(x)$
+ -->
+<IMG
+ WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img405.gif"
+ ALT="$ E(x), H(x)$"> are complex-valued vector fields and <!-- MATH
+ $\omega\neq 0$
+ -->
+<IMG
+ WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img406.gif"
+ ALT="$ \omega\neq 0$"> is a given angular frequency. 
+</FONT><P>
+<DIV><B>R<SMALL>EMARK</SMALL>  8</B> &nbsp; 
+For example, a monofrequent laser can be described by the time-harmonic Maxwell's equations.</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">In the time-harmonic case the space and time variables decouple and we can eliminate the time dependency. For this, we ask 
+<!-- MATH
+ $E(x) \exp(i\omega t)$
+ -->
+<IMG
+ WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img407.gif"
+ ALT="$ E(x) \exp(i\omega t)$"> and <!-- MATH
+ $H(x) \exp(i\omega t)$
+ -->
+<IMG
+ WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img408.gif"
+ ALT="$ H(x) \exp(i\omega t)$"> to satiesfy (<A HREF="node6.html#eq:_maxwell">35</A>).
+By then inserting the second equation of (<A HREF="node6.html#eq:_maxwell">35</A>) into the first one, we can eliminate the magnetic field <IMG
+ WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img409.gif"
+ ALT="$ H(x)$">. This yields
+</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="284" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img410.gif"
+ ALT="$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0
+$">
+</DIV><P></P><FONT SIZE="-1">
+In the <I>low-frequency case</I> where <IMG
+ WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img411.gif"
+ ALT="$ \vert\omega\vert$"> is small, it is known that for general materials the material parameters are such that
+</FONT><!-- MATH
+ \begin{displaymath}
+\omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="201" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img412.gif"
+ ALT="$\displaystyle \omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+Hence, neglecting the expression <!-- MATH
+ $\omega^2\varepsilon E(x)$
+ -->
+<IMG
+ WIDTH="71" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img413.gif"
+ ALT="$ \omega^2\varepsilon E(x)$"> is reasonable and it brings us to the low-frequency approximation of the
+time-harmonic Maxwell's equations:
+</FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="222" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img414.gif"
+ ALT="$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0
+$">
+</DIV><P></P>
+<P>
+<FONT SIZE="-1">We consider this equation in a conductor <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $"> (<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img398.gif"
+ ALT="$ \sigma(x)$"> pos.&nbsp;def.) and a impose Dirichlet boundary condition on the tangential trace
+of the field:
+</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_non-hom._bc"></A><!-- MATH
+ \begin{equation}
+E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="170" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img415.gif"
+ ALT="$\displaystyle E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(36)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+Proceeding as in [<A
+ HREF="node4.html#Alonso-Valli">1</A>], we assume that a vector funciton <IMG
+ WIDTH="20" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img416.gif"
+ ALT="$ \tilde{E}$"> is known, 
+satiesfying (<A HREF="node6.html#eq:_non-hom._bc">36</A>), and we end up with the following boundary value problem for <!-- MATH
+ $\underline u = E - \tilde{E}$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img417.gif"
+ ALT="$ \underline u = E - \tilde{E}$">
+</FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_real_problem"></A><!-- MATH
+ \begin{equation}
+\begin{split}
+\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) + i\omega\sigma u &= F \quad \mathrm{in} \quad \Omega \,, \\
+	u \wedge n &= 0 \quad \mathrm{on} \quad \partial \Omega \,.
+\end{split}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="426" HEIGHT="55" BORDER="0"
+ SRC="img418.gif"
+ ALT="\begin{displaymath}\begin{split}\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) +...
+... n &amp;= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(37)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+Although problem (<A HREF="node6.html#eq:_real_problem">37</A>) is complex-valued, finding a finite element method to approximate (<A HREF="node6.html#eq:_real_problem">37</A>)
+basically boils down to finding a finite element method for the real valued model problem (<A HREF="node1.html#eq:model_problem">1</A>).
+</FONT>
+<P>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
diff --git a/deal.II/doc/toc.html b/deal.II/doc/toc.html
index 0397686d23..2403d584ec 100644
--- a/deal.II/doc/toc.html
+++ b/deal.II/doc/toc.html
@@ -204,7 +204,8 @@
         &nbsp; MinRes linear solver.
 
     <li><em>Anna Schneebeli:</em>
-	Help and advice for Nedelec elements.
+	Help and advice for Nedelec elements, writing the excellent
+	report on Nedelec elements.
 
     <li><em>Jan Schrage:</em>
 	Initial parts of the tutorial.