From 9de6ec425d94852eb451644c2160d2e2fe0fe7f8 Mon Sep 17 00:00:00 2001 From: wolf Date: Wed, 30 Apr 2003 22:27:55 +0000 Subject: [PATCH] Add Annas Nedelec report. git-svn-id: https://svn.dealii.org/trunk@7519 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/doc/documentation.html | 14 + deal.II/doc/reports/nedelec/images.aux | 2 + deal.II/doc/reports/nedelec/images.bbl | 78 + deal.II/doc/reports/nedelec/images.log | 2433 ++++++++ deal.II/doc/reports/nedelec/images.tex | 2830 +++++++++ deal.II/doc/reports/nedelec/img1.gif | Bin 0 -> 1981 bytes deal.II/doc/reports/nedelec/img10.gif | Bin 0 -> 267 bytes deal.II/doc/reports/nedelec/img100.gif | Bin 0 -> 7258 bytes deal.II/doc/reports/nedelec/img101.gif | Bin 0 -> 21672 bytes deal.II/doc/reports/nedelec/img102.gif | Bin 0 -> 429 bytes deal.II/doc/reports/nedelec/img103.gif | Bin 0 -> 11823 bytes deal.II/doc/reports/nedelec/img104.gif | Bin 0 -> 6618 bytes deal.II/doc/reports/nedelec/img105.gif | Bin 0 -> 1187 bytes 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deal.II/doc/reports/nedelec/main.css create mode 100644 deal.II/doc/reports/nedelec/main.html create mode 100644 deal.II/doc/reports/nedelec/nedelec.ps create mode 100644 deal.II/doc/reports/nedelec/node1.html create mode 100644 deal.II/doc/reports/nedelec/node2.html create mode 100644 deal.II/doc/reports/nedelec/node3.html create mode 100644 deal.II/doc/reports/nedelec/node4.html create mode 100644 deal.II/doc/reports/nedelec/node5.html create mode 100644 deal.II/doc/reports/nedelec/node6.html 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 very detailed report on + Nedelec elements (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 printable version of the report. +

    + 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} diff --git a/deal.II/doc/reports/nedelec/images.log b/deal.II/doc/reports/nedelec/images.log new file mode 100644 index 0000000000..d036f77a48 --- /dev/null +++ b/deal.II/doc/reports/nedelec/images.log @@ -0,0 +1,2433 @@ +This is TeX, Version 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+\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% +\providecommand{\Z}{\mathbb{Z}}% +\providecommand{\C}{\mathbb{C}}% +\providecommand{\N}{\mathbb{N}}% +\providecommand{\inR}[1]{\in \mathbb{R}^{#1}}% +\providecommand{\EE}[1]{\mathbb{E}\,#1}%% mathematical expectation% +\providecommand{\PP}[1]{\mathbb{P}\,#1}%% mathematical probability% +\providecommand{\Or}[2]{\mathcal{O}(#1^#2)}%% order% +\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}% +\providecommand{\Section}[1]{\section{\large{#1}}}% +\providecommand{\SectionS}[1]{\section*{\large{#1}}}% +\providecommand{\Subsection}[1]{\subsection{\normalsize{#1}}}% +\providecommand{\SubsectionS}[1]{\subsection*{\normalsize{#1}}}% +\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{} + + +\usepackage[dvips]{color} + + +\pagecolor[gray]{.7} + 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+\newcommand\lthtmldisplayB[1]{\edef\preveqno{(\theequation)}% + \lthtmldisplayA{#1}\let\@eqnnum\relax}% +\newcommand\lthtmldisplayZ{\lthtmlboxmathZ\lthtmllogmath\lthtmlsetmath}% +\newcommand\lthtmlinlinemathA{\bgroup\catcode`\_=8 \lthtmlinlinemathB} +\newcommand\lthtmlinlinemathB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA + \vrule height1.5ex width0pt }% +\newcommand\lthtmlinlineA{\bgroup\catcode`\_=8 \lthtmlinlineB}% +\newcommand\lthtmlinlineB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA}% +\newcommand\lthtmlinlineZ{\egroup\expandafter\ifdim\dp\sizebox>0pt % + \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetinline} +\newcommand\lthtmlinlinemathZ{\egroup\expandafter\ifdim\dp\sizebox>0pt % + \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetmath} +\newcommand\lthtmlindisplaymathZ{\egroup % + \centerinlinemath\lthtmllogmath\lthtmlsetmath} +\def\lthtmlsetinline{\hbox{\vrule width.1em \vtop{\vbox{% + \kern.1em\copy\sizebox}\ifdim\dp\sizebox>0pt\kern.1em\else\kern.3pt\fi + \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}} +\def\lthtmlsetmath{\hbox{\vrule width.1em\kern-.05em\vtop{\vbox{% + \kern.1em\kern0.8 pt\hbox{\hglue.17em\copy\sizebox\hglue0.8 pt}}\kern.3pt% + \ifdim\dp\sizebox>0pt\kern.1em\fi \kern0.8 pt% + \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}} +\def\centerinlinemath{% + \dimen1=\ifdim\ht\sizebox<\dp\sizebox \dp\sizebox\else\ht\sizebox\fi + \advance\dimen1by.5pt \vrule width0pt height\dimen1 depth\dimen1 + \dp\sizebox=\dimen1\ht\sizebox=\dimen1\relax} + +\def\lthtmlcheckvsize{\ifdim\ht\sizebox<\vsize + \ifdim\wd\sizebox<\hsize\expandafter\hfill\fi \expandafter\vfill + \else\expandafter\vss\fi}% +\providecommand{\selectlanguage}[1]{}% +\makeatletter \tracingstats = 1 +\providecommand{\Eta}{\textrm{H}} +\providecommand{\Mu}{\textrm{M}} +\providecommand{\Alpha}{\textrm{A}} +\providecommand{\Iota}{\textrm{J}} +\providecommand{\Nu}{\textrm{N}} +\providecommand{\Omicron}{\textrm{O}} +\providecommand{\omicron}{\textrm{o}} +\providecommand{\Chi}{\textrm{X}} +\providecommand{\Beta}{\textrm{B}} +\providecommand{\Kappa}{\textrm{K}} +\providecommand{\Tau}{\textrm{T}} +\providecommand{\Epsilon}{\textrm{E}} +\providecommand{\Zeta}{\textrm{Z}} +\providecommand{\Rho}{\textrm{R}} + + +\begin{document} +\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength hsize=\the\hsize}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength vsize=\the\vsize}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength hoffset=\the\hoffset}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength voffset=\the\voffset}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength topmargin=\the\topmargin}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength topskip=\the\topskip}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength headheight=\the\headheight}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength headsep=\the\headsep}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength parskip=\the\parskip}\lthtmltypeout{}% +\lthtmltypeout{latex2htmlLength oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}% +\makeatletter +\if@twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}% +\else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi% +\lthtmltypeout{}% +\makeatother +\setcounter{page}{1} +\onecolumn + +% !!! IMAGES START HERE !!! + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5220}% +$ H(\mathop {\rm curl};\Omega )$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5236}% +$ L^2(\Omega )$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5250}% +$ H(\mathop {\rm curl};(\Omega ))$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +\stepcounter{section} +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5451}% +$ \Omega \in \mathbb{R}^d$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5453}% +$ d=2,3$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_indisplay5455}% +$\displaystyle \mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u = \underline f \quad \mathrm{in} \quad \Omega \,,$% +\lthtmlindisplaymathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5457}% +$ \underline f \in L^2(\Omega )^d$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_indisplay5459}% +$\displaystyle \underline u \wedge \underline n = 0$% +\lthtmlindisplaymathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5461}% +$ \partial \Omega $% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5463}% +$ \Omega $% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5465}% +$ c(x)$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +\stepcounter{subsection} +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5469}% +$ \underline t$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5471}% +$ d=2$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5473}% +$ \underline v = \left(\begin{array}{c} v_1(x,y) \\ v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmlinlinemathA{tex2html_wrap_inline5475}% +$ \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}$% +\lthtmlinlinemathZ +\lthtmlcheckvsize\clearpage} + +{\newpage\clearpage +\lthtmldisplayA{displaymath7248}% +\begin{displaymath}\begin{split} \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) + i\omega\sigma u &= F \quad \mathrm{in} \quad \Omega \,, \\ u \wedge n &= 0 \quad \mathrm{on} \quad \partial \Omega \,. 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    An +$ H(\mathop {\rm curl};\Omega )$-conforming FEM: +
    +Nédélec's elements of first type

    +

    Anna Schneebeli, April 30, 2003

    +

    +Abstract: The aim of this report is to give an introduction to Nédélec's +$ H(\mathop {\rm curl};\Omega )$-conforming finite element method of first +type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [    8]. +
    +In the first section, we present the model problem and introduce the framework for its variational formulation. +
    +In the second section, we present Nédélec's elements of first type for +$ 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. +
    +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. +
    +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. +     +

    + +

    + + + + + +
    +
    + +2003-04-30 +
    + + diff --git a/deal.II/doc/reports/nedelec/main.css b/deal.II/doc/reports/nedelec/main.css new file mode 100644 index 0000000000..aad8978165 --- /dev/null +++ b/deal.II/doc/reports/nedelec/main.css @@ -0,0 +1,30 @@ +/* Century Schoolbook font is very similar to Computer Modern Math: cmmi */ +.MATH { font-family: "Century Schoolbook", serif; } +.MATH I { font-family: "Century Schoolbook", serif; font-shape: italic } +.BOLDMATH { font-family: "Century Schoolbook", serif; font-weight: bold } + +/* implement both fixed-size and relative sizes */ +SMALL.XTINY { font-size : xx-small } +SMALL.TINY { font-size : x-small } +SMALL.SCRIPTSIZE { font-size : smaller } +SMALL.FOOTNOTESIZE { font-size : small } +SMALL.SMALL { } +BIG.LARGE { } +BIG.XLARGE { font-size : large } +BIG.XXLARGE { font-size : x-large } +BIG.HUGE { font-size : larger } +BIG.XHUGE { font-size : xx-large } + +/* heading styles */ +H1 { } +H2 { } +H3 { } +H4 { } +H5 { } + +/* mathematics styles */ +DIV.displaymath { } /* math displays */ +TD.eqno { } /* equation-number cells */ + + +/* document-specific styles come next */ diff --git a/deal.II/doc/reports/nedelec/main.html b/deal.II/doc/reports/nedelec/main.html new file mode 100644 index 0000000000..c7412e06ac --- /dev/null +++ b/deal.II/doc/reports/nedelec/main.html @@ -0,0 +1,157 @@ + + + + + +An -conforming FEM: Nédélec's elements of first type + + + + + + + + + + + + + + +

    An +$ H(\mathop {\rm curl};\Omega )$-conforming FEM: +
    +Nédélec's elements of first type

    +

    Anna Schneebeli, April 30, 2003

    +

    +Abstract: The aim of this report is to give an introduction to Nédélec's +$ H(\mathop {\rm curl};\Omega )$-conforming finite element method of first +type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [8]. +
    +In the first section, we present the model problem and introduce the framework for its variational formulation. +
    +In the second section, we present Nédélec's elements of first type for +$ 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. +
    +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. +
    +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. +     +

    + +

    + + + + + +
    +
    + +2003-04-30 +
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w(0.99)p 2578 2166 V 102 w(4.01e-02)p +3001 2166 V 103 w(1.00)p 3262 2166 V 725 2170 2540 4 +v 28 2436 a(T)-8 b(able)40 b(1:)63 b(Errors)40 b(and)h(con)m(v)m +(ergence)i(rates)f(in)e(the)h Fc(L)1993 2403 y Fb(2)2032 +2436 y Fk(\(\012\)-norm)h(and)e Fc(H)7 b Fk(\(curl)o(;)15 +b(\012\)-seminorm)41 b(for)f(the)i(t)m(w)m(o)28 2549 +y Fa(MATLAB)28 b Fk(examples.)872 4420 y @beginspecial +52 @llx 194 @lly 549 @urx 605 @ury 2693 @rwi 1984 @rhi +@setspecial +%%BeginDocument: example1_errors.eps +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The Mathworks, Inc. +%%Title: examlpe1_error.eps +%%CreationDate: 08/26/2002 11:49:36 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Pages: 1 +%%BoundingBox: 52 194 549 605 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s /show ldef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke + cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup string + currentfile exch readhexstring pop + /bmap xdef pop pop} bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + {bmap} image gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup 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+eplot +%%EndObject + +epage +end + +showpage + +%%Trailer +%%EOF + +%%EndDocument + @endspecial 28 4616 a(Figure)f(1:)39 b(Con)m(v)m(ergence)30 +b(of)d(the)h(FE-appro)m(ximation)f(to)i(the)e(smo)s(oth)h(solution)e +(of)i(the)f Fa(MATLAB)f Fk(example)i(\(31\))28 4729 y(in)h(the)h +Fc(L)352 4696 y Fb(2)392 4729 y Fk(\(\012\)-norm)g(and)g(the)g +Fc(H)7 b Fk(\(curl)o(;)15 b(\(\012\)\)-seminorm)28 5150 +y Fn(Remark)28 b Fz(7)45 b Fm(The)22 b(mesh)g(gener)l(ation)f(and)h(r)l +(e\014nement)e(was)i(done)g(by)g Fd(PDE-toolbox)16 b +Fm(c)l(ommands.)37 b(Sinc)l(e)21 b(the)g Fd(PDE-toolbox)28 +5250 y Fm(do)l(es)32 b(not)f(supp)l(ort)h(thr)l(e)l(e)g(dimensional)h +(grids,)h(we)e(r)l(estricte)l(d)g(ourselves)g(to)g(2d)g(pr)l(oblems,)i +(and)e(we)g(have)h(so)f(far)h(no)e(nu-)28 5349 y(meric)l(al)f(r)l +(esults)f(for)i(the)f(c)l(ase)g(of)h(tetr)l(ahe)l(dr)l(al)f(grids)h(in) +f(3d.)1949 5719 y Fk(17)p eop +%%Page: 18 18 +18 17 bop 872 1803 a @beginspecial 52 @llx 194 @lly 549 +@urx 605 @ury 2693 @rwi 1984 @rhi @setspecial +%%BeginDocument: example2_errors.eps +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The Mathworks, Inc. +%%Title: pix.eps +%%CreationDate: 08/26/2002 11:52:50 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Pages: 1 +%%BoundingBox: 52 194 549 605 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s /show ldef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke + cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup string + currentfile exch readhexstring pop + /bmap xdef pop pop} bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + {bmap} image gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 52 194 549 605 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 12 def +portraitMode 0204 7344 csm + + 424 80 5969 4933 MR c np +92 dict begin %Colortable dictionary +/c0 { 0 0 0 sr} bdef +/c1 { 1 1 1 sr} bdef +/c2 { 1 0 0 sr} bdef +/c3 { 0 1 0 sr} bdef +/c4 { 0 0 1 sr} bdef +/c5 { 1 1 0 sr} bdef +/c6 { 1 0 1 sr} bdef +/c7 { 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549 611 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s /show ldef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw 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grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke + cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup string + currentfile exch readhexstring pop + /bmap xdef pop pop} bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + {bmap} image gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 55 201 549 611 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 12 def +portraitMode 0204 7344 csm + + 457 4 5930 4927 MR c np +92 dict begin %Colortable dictionary +/c0 { 0 0 0 sr} bdef +/c1 { 1 1 1 sr} bdef +/c2 { 1 0 0 sr} bdef +/c3 { 0 1 0 sr} bdef +/c4 { 0 0 1 sr} bdef +/c5 { 1 1 0 sr} bdef +/c6 { 1 0 1 sr} bdef +/c7 { 0 1 1 sr} bdef +c0 +1 j +1 sg + 0 0 6913 5185 PR +6 w +0 4225 5356 0 0 -4225 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mode 100644 index 0000000000..169abda394 --- /dev/null +++ b/deal.II/doc/reports/nedelec/node1.html @@ -0,0 +1,1018 @@ + + + + + +1 The model problem and the space + + + + + + + + + + + + + + + +Subsections + + + +
    + +

    +1 The model problem and the space +$ H(\mathop {\rm curl};\Omega )$ +

    +

    +Consider the vector-valued model problem in a Lipschitz domain +$ \Omega \in \mathbb{R}^d$, $ d=2,3$: +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u = \underline f \quad \mathrm{in} \quad \Omega \,,$ +(1)
    +

    +with right hand side +$ \underline f \in L^2(\Omega )^d$. +
    +We assume a homogeneous Dirichlet boundary condition on the tangential trace +

    +
    + + + + +
    $\displaystyle \underline u \wedge \underline n = 0$ +(2)
    +

    +on the boundary +$ \partial \Omega $ of $ \Omega $. +
    +The coefficient $ c(x)$ is assumed to be bounded and uniform positive definite. +
    +This type of problem typically arises in particular settings of +Maxwell`s equations. The boundary condition (2) then applies to a perfectly conducting boundary. +For a derivation of the model problem (1), refer to Appendix A. +
     +

    +The subject of this section is to give an appropriate setting for a variational formulation of (1). +
    +A more detailed treatment of the following notions and proofs can be found in [4]. +     +

    + +

    +1.1 Definitions +

    +

    +

    +

    CONVENTION 1   +In the following, the vector +$ \underline t$ will denote the unit tangent vector w. r. 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" ).

    + +

    +Let us first consider the case of $ d=2$. For +$ \underline v = \left(\begin{array}{c} v_1(x,y) \\ v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$ +and +$ \varphi \in \mathcal{D}(\overline{\Omega })$ we define the scalar- and the +vector-valued curl-operators: + +

    +
    +$\displaystyle \mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_... +...in{array}{c} \partial _y\varphi \\ -\partial _x\varphi \end{array}\right) \,. +$ +

    +We note that the +$ \mathop{\rm curl}\mathop{\rm curl}$-operator in two dimensions has to be understood as +$ \mathop{\underline{\rm curl}}\mathop{\rm curl}$. +

    +

    REMARK 1   +In the two dimensional case, the +$ \mathop{\rm curl}$ operator is simply the divergence of the rotated field +$ \underline v$. Similarly, the +$ \mathop{\underline{\rm curl}}$ operator is +the rotated gradient field of $ \varphi $. Setting + +

    +
    +$\displaystyle \boldsymbol{R} = \left(\begin{array}{cc} +0 & 1 \\ +-1 & 0 +\end{array}\right) \,, +$ +

    +we have + +

    +
    +$\displaystyle \mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right) +$ +

    +and + +

    +
    +$\displaystyle \mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,. +$ +

    +We further note that the tangential vector +$ \underline t$ is just the rotated outward unit normal vector + +$ \underline t = \boldsymbol{R}^T\underline n$. This will enable us to derive statements for the +$ \mathop{\rm curl}$-operators in two dimensions from statements for the +divergence and gradient operators in two dimensions.

    + +

    +For the case of $ d=3$ and a vector field +$ \underline v \in [\mathcal{D}(\overline{\Omega })]^3$ we write + +

    +
    +$\displaystyle \mathop{\rm curl}\underline v := \nabla \wedge \underline v := \l... +... - \partial _x v_3 \\ +\partial _x v_2 - \partial _y v_1 +\end{array} \right) +$ +

    +

    DEFINITION 1   +For $ d=2,3$ we write +$ \tilde{d}=1$ if $ d=2$ and +$ \tilde{d}=3$ if $ d=3$, and we define + +

    +
    +$\displaystyle H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d: \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \} +$ +

    + + +$ H(\mathop {\rm curl};\Omega )$ endowed with the inner product + +

    +
    +$\displaystyle (\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\... +... (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )} +$ +

    +is a Hilbert space. +
     +

    + +

    +1.2 Trace theorem, integration by parts +

    +The space +$ 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 +$ H(\mathop {\rm curl};\Omega )$-function onto the boundary +$ \partial \Omega $ and we define intergation by parts on the +space +$ H(\mathop {\rm curl};\Omega )$. + +

    +

    +

    THEOREM 1 (Approximation Property)   +For $ d=2,3$, +$ [\mathcal{D}(\overline{\Omega })]^d$ is dense in +$ H(\mathop {\rm curl};\Omega )$.

    + +See [4] 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 & Lions, 1971. +
     +

    +Equipped with this approximation property of smooth functions to elements of +$ H(\mathop {\rm curl};\Omega )$, we can state + +

    +

    +

    THEOREM 2 (Green's Formula)   +For the 2d case, let +$ \underline u$ be in +$ [H(\mathop{\rm curl};\Omega )]^2$ and $ \varphi $ be a test function in +$ H^1(\Omega )$. We have + +

    +
    +$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx = \... +...+ \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:, +$ +

    + +

    +For the 3d case, let +$ \underline u$ be in +$ [H(\mathop{\rm curl};\Omega )]^3$ and +$ \underline v$ be a test function in +$ [H^1(\Omega )]^3$. We then have + +

    +
    +$\displaystyle \int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \... +...\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:, +$ +

    +The boundary integrals are understood as duality pairings in +$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$ . +

    +PROOF. 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 1. +
    +For the 3d case we use the identity +     +

    +
    +$\displaystyle \mathrm{div}\, (\underline u \wedge \underline v) = \underline v ... +...thop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v +$ +

    +together with Gauss' Divergence Theorem and the properties of the mixed product +$ (\underline a\wedge\underline b)\cdot \underline c$ to obtain + +

    +
    +$\displaystyle \int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - ... +...rtial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,. +$ +

    + The extention to a pairing of + +$ H(\mathop{\rm curl})$ and +$ H^1(\Omega )$ functions follows with Theorem 1 by a density argument and is a result of the proof of the + Trace Theorem. See [4] p.21 for details. + + +

    +

    +

    THEOREM 3 (Trace Theorem)   +For $ d=3$, let +$ \underline n$ denote the outward unit normal to the boundary +$ \partial \Omega $. For $ d=2$, let +$ \underline t$ be as in convention 1 +
    +For $ d=2$ the mapping + +

    +
    +$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t +$ +

    +and for $ d=3$ the mapping + +

    +
    +$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n +$ +

    +is contiuous and linear from +$ H(\mathop {\rm curl};\Omega )$ to +$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$.

    + +

    +Note, that the trace of a +$ H(\mathop {\rm curl};\Omega )$-function is only defined in tangential direction. Its trace is in the dual space + of traces of +$ [H^1(\Omega )]^{\tilde{d}}$ functions. Recall that traces of such functions +are defined in every direction and are functions in +$ [H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$. +
     +

    +PROOF. +The proof of the trace theorem follows from Green's formula stated in theorem 2 applied to smooth functions +and then by density arguments. See [4] p.21 for details. + + +

    +Due to the Trace Theorem it makes sense to define a space of +$ H(\mathop{\rm curl})$-functions with vanishing tangential components on the boundary. +

    +

    DEFINITION 2   + +

    +
    +$\displaystyle H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mat... +...nderline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\} +$ +

    + +

    +

    +

    REMARK 2   +For $ d=2,3$, +$ [\mathcal{D}(\Omega )]^d$ is dense in +$ H_0(\mathop{\rm curl};\Omega )$. +

    + +

    +A consequence of Green's formula is the following important regularity property of +$ H(\mathop {\rm curl};\Omega )$-functions: + +

    +

    +

    PROPOSITION 1   +Let $ K_-$ and $ K_+$ be two polygonal (resp. polyhedral) Lipschitz domains in +$ \mathbb{R}^d$, with a common edge (resp. common edge or face) + +$ e = \partial K_-\cap\partial K_+ \neq \emptyset$ and + denote by +$ \Omega = \partial K_-\cup\partial K_+$ their union. A function $ v$ is in +$ H(\mathop {\rm curl};\Omega )$ if and only if the restricion $ v_-$ of $ v$ to $ K_-$ + is in +$ H(\mathop{\rm curl}; K_-)$, the restricion $ v_+$ of $ v$ to $ K_+$ is in +$ H(\mathop{\rm curl}; K_+)$ and the tangential jump over $ e$ vanishes: +$ v_-\wedge n_- + v_+\wedge n_+ = 0$ on $ e$. +

    + + PROOF. + The proposition follows from choosing an appropriate test function and integrating by parts (global and local). + In order to localise the result of the Trace Theorem, we must choose a testfunction from the space +$ H^{\frac{1}{2}}_{00}(e)$. These + functions vanish at the endpoints of $ e$ and can therefore be extended by zero to a +$ H^{\frac{1}{2}}(\partial \Omega )$-function. From the comparison + of local (on $ K_-$ and $ K_+$ separately) and global (on $ \Omega $) integration by parts it followas then that the tangential jump vanishes in the + dual space of +$ H^{\frac{1}{2}}_{00}(e)$. By densitiy properties of +$ H^{\frac{1}{2}}_{00}(e)$ it follows that the tangential traces vanish in + the "correct space" as well. The "correct space" would be +$ H^{-\frac{1}{2}}(e)$ if we have no further regularity of +$ \underline v_-$ and +$ \underline v_+$, and it + would be $ L^2(e)$ if +$ \underline v$ is elementwise in $ H^1$ (e. g.  for piecewise polynomial +$ \underline v$). + + +

    + +

    +1.3 Variational formulation of the model problem +

    +In the previous sections we introduced the space +$ H(\mathop {\rm curl};\Omega )$, an integration-by-parts formula and the notion of trace for an + +$ H(\mathop {\rm curl};\Omega )$-function. In this framework, the variational formulation of the model problem (1) reads: +
    +
    +
    Find +$ \underline u \in H_0(\mathop{\rm curl};\Omega )$ such that for all test functions +$ \underline v \in H_0(\mathop{\rm curl};\Omega )$ holds +

    +
    + + + + +
    $\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\... +..., \cdot \underline v\,dx = \int_{\Omega } \underline f\, \cdot \underline v\,dx$ +(3)
    +

    +
    +
    +With our assumptions on the data, the forms +

    +
    + + + + +
    \begin{displaymath}\begin{split}a(\underline u,\underline v) &:= \int_{\Omega } ... +...int_{\Omega } \underline f\, \cdot \underline v\,dx \end{split}\end{displaymath} +   
    +

    +are continuous and the bilinear form +$ a(\cdot,\cdot)$ is coercive on + +$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$. By the Lax-Milgram lemma it follows, that there exists a unique solution + +$ \underline u \in H_0(\mathop{\rm curl};\Omega )$ of (3). + +
    +
    + +2003-04-30 +
    + + 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 @@ + + + + + +2 Nédélec's elements of first type for + + + + + + + + + + + + + + + +Subsections + + + +
    + +

    +2 Nédélec's elements of first type for +$ H(\mathop {\rm curl};\Omega )$ +

    + In this section we will present present +$ H(\mathop {\rm curl};\Omega )$-conforming vector-valued finite elements, + the Nédélec elements of first type (cf. [8]), which can be used to discretize the variational problem (3). +
     +

    +In order to define a finite element we must specify +

    +
    the geometry
    +
    We choose a reference element $ \hat{K}$ and a change of variables +$ F_K(\hat{x})$, the element map. + We set +$ K = F_K(\hat{K})$. + +
    +
    a function space
    +
    We need a finite dimensional function space $ \hat{R}$, typically a space of polynomials, on the reference + cell, plus a transformation of $ \hat{R}$ to a function space $ R_K$ on a general cell $ K$. + +
    +
    dofs
    +
    We have to define a set of dofs +$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$. These are linear functionals on + $ \hat{R}$ and +$ N < \infty$ is the dimension of $ \hat{R}$. +$ \mathcal{A}$ should be unisolvent, that is, the dofs +$ \alpha_i(\cdot)$ + are linearly independent. + +
    +
    +

    +First, we observe that for a conforming discretization of (3) 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 +$ 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). +
    +We will give an outline of the construction of the finite element spaces described by Nédélec in [8]. In literature, they are also referred to as + Nédélec's elements of first type. +
     +

    + +

    +2.1 Construction of Nédélec elements on tetrahedral grids +

    +

    +In this section, we denote by $ \hat{K}$ the standard triangular or tetrahedral reference element. + +

    + +

    +2.1.1 Polynomial spaces on the reference element +

    + In [8], Nédélec introduces the function spaces +$ \hat{R} = \mathcal{R}^k$, on which his finite element will be based. These spaces are subject to this + section. + For more details, consult [8]. +
    +We denote by +$ \mathbb{P}_k(\hat{\Sigma})$ the space of polynomials of degree $ k$ on +$ \hat{\Sigma}$, where +$ \hat{\Sigma}$ is an edge, a + face of or the reference element itself.. The space +$ \tilde{\mathbb{P}}_k$ of homogeneous + polynomials of degree $ k$ is the span of monomials of total degree $ k$ in $ d$ variables on $ \hat{K}$. +     +

    +

    +

    DEFINITION 3   +We define the auxiliary space +

    +
    + + + + +
    $\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d :... +...ine p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$ +(4)
    +

    +with +$ \hat{x} \in \hat{K}$. +
    +The dimension of this space is $ k$ in the case $ d=2$ and $ k(k+2)$ for $ d=3$. +

    + +

    +Nédélec's first family of +$ H(\mathop {\rm curl};\Omega )$-conforming finite elements is based on the polynomial spaces +

    +

    DEFINITION 4   +

    +
    + + + + +
    $\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$ +(5)
    +

    +These spaces have dimension +

    +
    + + + + +
    \begin{displaymath}\begin{split}\mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \t... +...ac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. \end{split}\end{displaymath} +   
    +

    + +

    +In the two-dimensional case, an equivalent characterization of the space +$ \mathcal{R}^k$ is + +

    +

    +
    + + + + +
    $\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus ... +..._{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right)\,.$ +(6)
    +

    + This can be seen by noting that for $ d=2$ + +

    +
    +$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ ... +...ilde{p} \left(\begin{array}{cc} x_2 \\ -x_1 \end{array}\right)\,,\, \tilde{p} +$ +

    + obviously holds. + Moreover, the dimension of the space +$ \tilde{\mathbb{P}}_{k-1}$ of homogeneous polynomials of degree $ k-1$ in two variables is $ k$ and this is + also the dimension +$ \mathcal{S}^k$. This proves the stated equivalent representation of the space +$ \mathcal{S}^k$. +
    +
    +We illustrate these definitions with some examples. We start with the case $ d=2$ and consider the spaces of polynomials of degree + $ k=1$ and $ k=2$: +     +

    +

    +

    EXAMPLE 1   + +

    +

    +
    + + + + +
    $\displaystyle \mathcal{R}^1 = \left\langle \left(\begin{array}{cc} 1 \\ 0 \end... +...eft(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right) \right\rangle$ +(7)
    +

    + +

    + +

    +
    +$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus +\... +...{cc} {\hat{x}_2}^2 \\ -\hat{x}_1\,\hat{x}_2 \end{array}\right) +\right\rangle +$ +

    + +

    +

    + +

    +To illustrate a case for $ d=3$, we consider the lowest polynomial degree $ k=1$: +

    +

    EXAMPLE 2   +We have to specify a basis for +$ \mathcal{S}^1$: +
    +Let +$ \underline p$ be a polynomial in +$ (\mathbb{P}_{1}(\hat{K}))^3$ with componentwise representation + +

    +
    +$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,. +$ +

    +The condition for +$ \underline p$ being in +$ \mathcal{S}^1$ is + +

    +
    +$\displaystyle \underline p \cdot \hat{\underline x} = +\sum_{i=1}^3 a_{ii}\hat{... +...um_{\substack{i,j=1 \\ j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,. +$ +

    +This leads to the condition on the coefficients of a polynomial in +$ \mathcal{S}^1$: +

    +
    + + + + +
    \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} +   
    +

    + +

    +With the basis of +$ \mathcal{S}^1$ which is obtained by choosing +$ a_{ij} = 1$, $ i=1,2,3$, $ j>i$ and setting all the other + coefficients to zero, we get +

    +
    + + + + +
    $\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus \l... +...egin{array}{ccc} \hat{x}_2 \\ \hat{x}_1 \\ 0 \end{array}\right) \right\rangle$ +   
    +

    + +

    +

    + +

    +We remark at this point that the spaces +$ \mathcal{R}^k$ do not span the whole +$ (\mathbb{P}_{k}(\hat{K}) )^d$. + An +$ H(\mathop {\rm curl};\Omega )$-conforming FEM based on full polynomial spaces, the so called + Nédélec elements of second type, was introduced in 1986 by Nédélec in [10]. + +

    +

    +

    REMARK 3   +The original, rather technical, representation of the spaces +$ \mathcal{R}^k$ is given in Definition 2 in [8]. + Nédélec uses this representation in most of his proofs. We will not refer to it here. +

    + +

    + +

    + +
    +2.1.2 Degrees of freedom on the reference element +

    + In this section we define the set +$ \mathcal{A}$ of dofs, which is a set of linear functionals on +$ \mathcal{R}^k$. +

    +

    REMARK 4   +Recall that the dimension of the spaces of polynomials of degree $ k$ in $ n$ variables is +$ n+k+2 \choose n$. +

    + +

    +

    +

    DEFINITION 5   +Let $ \hat{K}$ be the reference triangle and +$ \hat{\underline t}$ the tangent as defined in convention 1. + The set of degrees of freedom +$ \mathcal{A}$ on +$ \mathcal{R}^k$ in the 2d case consists of the linear functionals +
    +
    edge dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli... +...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$ +   
    +

    +for every edge $ \hat{e}$ of $ \hat{K}$. We have a total of $ 3k$ of edge dofs. + +

    +

    +
    inner dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin... +...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$ +   
    +

    +We have a total of $ k(k-1)$ of inner dofs. + +

    +

    +

    + +

    +

    +

    DEFINITION 6   +Let $ \hat{K}$ be the reference tetrahedron, +$ \hat{\underline t}$ the tangent to an edge as defined in convention 1 + and +$ \hat{\underline n}$ the outward unit normal vector to a face. + The set of degrees of freedom +$ \mathcal{A}$ on +$ \mathcal{R}^k$ in the 3d case consists of the linear functionals +
    +
    edge dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli... +...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$ +   
    +

    +for every edge $ \hat{e}$ of the tetrahedron $ \hat{K}$. We have a total of $ 6k$ of edge dofs. + +

    +

    +
    face dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli... +... }\,d\hat{a} \quad \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$ +   
    +

    +for every face $ \hat{f}$ of the tetrahedron $ \hat{K}$. We have a total of $ 4k(k-1)$ of face dofs. + +

    +

    +
    inner dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin... +...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$ +   
    +

    +We have a total of +$ \frac{k(k-1)(k-2)}{2}$ of inner dofs. + +
    +

    + +

    +We note that in the case of lowest order elements, i. e. $ k=1$, only edge dofs occur. This is not so for higher order elements. For + $ k=2$ we additionally have inner dofs in the 2d case and face dofs in the 3d case. For $ k\leq3$ we have all types of dofs in both + cases. +
    +We also note that the total number of dofs equals the dimension of the spaces +$ \mathcal{R}^k$, as it should be. +
    +The representation of the interface dofs, that is edge dofs in 2d, edge and face dofs in 3d, is motivated by the continuity + condition on +$ H(\mathop {\rm curl};\Omega )$-functions stated in proposition 1. +     +

    +

    +

    PROPOSITION 2   +The set +$ \mathcal{A}$ of dofs befined above is unisolvent on +$ \mathcal{R}^k$. + +$ \hat{\underline u}\in \mathcal{R}^k$ is uniquely defined by the moments +$ \hat{\alpha}(\hat{\underline u})$. +

    + + PROOF. See [8], proof of theorem 1 and preceeding lemmas. + +

    +

    +

    EXAMPLE 3 (Reference shape functions of lowest order for Nédélec elements on triangular meshes)   +Let the reference element be the triangle + +$ \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\}$. + Label the edges couterclockwise startung with +$ \hat{e}_0 = \overline{(0,0),(1,0)}$. + The tangential vectors to the edges are (oriented counterclockwise) + +

    +
    +$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array... +... +\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,. +$ +

    +The underlying function space for lowest order Nédélec elements on a triangular mesh is +$ \mathcal{R}^1$ from ([*]). +
    +In the case of $ k=1$ only egde-dofs occur. On $ \hat{K}$ we have dofs of the type + +$ \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)$. + More precisely, since +$ \varphi \equiv 1$ is a basis for +$ \mathbb{P}_{0}(\hat{e}_i)$ we have the three dofs +

    +
    + + + + +
    $\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$ +   
    +

    +In order to construct a FE-basis +$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$ for +$ \mathcal{R}^1$ with respect to these dofs, we require + +$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$. This leads to a linear system for + the coefficients of the +$ \hat{\underline N}_i$ in a general basis of +$ \mathcal{R}^1$. In the case of lowest order elements, it is easy to + verify that we have +

    +
    + + + + +
    $\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ \hat... +...ine N}_2 = \left(\begin{array}{cc} -\hat{y} \\ \hat{x}-1 \end{array}\right)\,.$ +(8)
    +

    + +

    + +

    +
    +2.1.3 Piola transformation +

    + An affine triangle or tetrahedron $ K$ is described by the affine element map + +

    +
    +$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K +$ +

    +

    +In standard +$ H^1(\Omega )$-conforming FEM, the shape functions $ N_i$ on a general cell $ K$ are obtained from the reference shape functions + $ \hat{N}_i$ on the reference element $ \hat{K}$ by the pull-back +

    +
    + + + + +
    $\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$ +   
    +

    + In the case of +$ H(\mathop {\rm curl};\Omega )$-conforming Nédélec FEM we cannot transforme our shape function in this way. The pull-back of a + +$ H(\mathop{\rm curl};\hat{K})$-function needs not to be in +$ H(\mathop{\rm curl}; K)$. In addition, the pull-back is not an +$ \mathcal{R}^k$-isomorphism and it + does not lead to an +$ H(\mathop {\rm curl};\Omega )$-conforming method if prescribing the dofs by definitions 5 or [*]. +
    +In Nédélec's FEM (or, more general, in +$ H(\mathop {\rm curl};\Omega )$-conforming FEM), the shape functions are transformed by the following covariant + transformation for vector-fields: +     +

    +

    +
    +
    The element shape functions +$ \underline N_i(x)$ on the element +$ K = F_K(\hat{K})$ are obtained + from the reference shape functions by +

    +
    + + + + +
    $\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$ +(9)
    +

    +where +$ \hat{D}F_K$ is the jacobian +$ \frac{d}{d\hat{x}}F_K(\hat{x})$ of the element map. + +
    +
    +

    +In literature, an equivalent to this transformation for +$ H(\mathop{\rm div}; \Omega )$-conforming FEM (which in that case is a contravariant map) + is referred to as Piola transformation, cf. [3] pp. 97. +
    +Here, we will refer to the transformation (9) of the vector field also as Piola transformation. +
    +We note that the gradients of scalar nodal +$ H^1(\Omega )$-conforming finite elements transform according to the Piola + transformation (9). +     +

    +In the case of tetrahedral elements and affine element map +$ F_K(\hat{x}) = B_K \hat{x} + b_k$, the jacobian +$ \hat{D}F_K$ + is just the constant matrix $ B_K$ and we have +

    +
    + + + + +
    $\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$ +(10)
    +

    +

    + +

    + +
    +2.1.4 Transformation of the curl in 2d +

    + For +$ \Omega \subset\mathbb{R}^2$, we noted in remark 1 that vector fields in +$ H(\mathop {\rm curl};\Omega )$ can be represented as + rotated +$ H(\mathop{\rm div}; \Omega )$ vector fields. + Moreover, it is easy to verify that +

    +
    + + + + +
    $\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$ +(11)
    +

    + where $ R$ is the rotation matrix from remark 1. + Therefore, the properties of the Piola transformation (10) in the 2d case can be derived directly + from the properties + of the +$ H(\mathop{\rm div}; \Omega )$-Piola transformation stated in [3] pp. 97. + +

    +

    +

    THEOREM 4 (Some properties of 2d Piola transformation for affine element map)   +Let +$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$, +$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$, +$ \hat{x} = +F_K^{-1}(x)$, with affine element map $ F_K$. +
    +
    (i)
    +
    The gradient +$ D\underline v$ transforms according to +

    +
    + + + + +
    $\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$ +(12)
    +

    + +
    +
    (ii)
    +
    The curl transforms according to +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$ +(13)
    +

    +As a consequence we see that +$ H(\mathop{\rm curl}; K)$ is isomorphic to +$ H(\mathop{\rm curl};\hat{K})$ under the Piola transformation + (10). + +
    +

    + +

    +PROOF. +

    +
    (i)
    +
    Chain rule + +
    +
    (ii)
    +
    We use that the 2d +$ \mathop{\rm curl}$ operator is just the trace of the rotated jacobian $ R\,Dv$. By remark 11, + we can replace $ B_K^{-T}$ and we get that $ R\,Dv$ is affine-equivalent to +$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$, + which proves (ii). + +
    +
    + + +

    +

    +

    COROLLARY 1   +From (ii) in theorem 4 we deduce +

    +
    + + + + +
    $\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\h... +...}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$ +   
    +

    +and we have, together with (ii) from theorem 4 +

    +
    + + + + +
    $\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underl... +...hat{\underline v}\, \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.$ +   
    +

    + +

    + +

    + +

    + +
    +2.1.5 Transformation of the curl in 3d +

    + 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. +
    +By the chain rule, we obtain the transformation of the gradient of a vector field +$ \underline v$, defined by the Piola transformation ([*]) of a reference field +$ \hat{\underline v}$: +

    +
    + + + + +
    $\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$ +(14)
    +

    +

    +We introduce the skew symmetric matrix +$ \mathop{\rm Curl}v$ as +

    +
    + + + + +
    $\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$ +(15)
    +

    +

    +We see that +$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$ and therefore by (14) +

    +
    + + + + +
    $\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$ +(16)
    +

    +

    +

    +

    PROPOSITION 3 (Transformation of the curl in 3d)   +Let $ \hat{K}$ be the reference tetrahedron and +$ K = F_K(\hat{K})$ an affine image of it. + The curl of a vector field +$ \underline v(x)$ on $ K$, defined by the Piola transformation of a reference field +$ \hat{\underline v}(\hat{x})$ + transforms according to +

    +
    + + + + +
    $\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$ +(17)
    +

    +We obtain the matrix +$ \mathrm{M_i}$ by replacing i-th column of the (constant) jacobian +$ D(F_K^{-1}) = B_K^{-1}$ + by the vector +$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$: +

    +
    + + + + +
    $\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll} (\... +...} \quad l=i \\ (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i \end{array} \right.$ +   
    +

    + (Note: an alternative, equivalent, transformation formula for the curl in 3d is given in proposition 4). +

    + +

    +PROOF. + It holds +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\math... +...\mathop{\rm Curl}v})_{31} \\ ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$ +(18)
    +

    + We demonstrate the statement of the proposition for the first component of the curl, which is +$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$. + Using the transformation (16), implicit summation over equal indices and the + abbreviation +$ b_{ij} := (B_K^{-1})_{ij}$, we have + +

    +
    +$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3} +$ +

    + Writing this out and recalling that +$ \mathop{\rm Curl}v$ is skew symmetric, yields + +

    +
    +$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\wide... +... ++(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,, +$ +

    + and with (18) this is equal to the determinant of + +

    +
    +$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm cu... +... +(\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33} +\end{array}\right) \,. +$ +

    + The proof for the other components follows analogously. + + +

    +In the next proposition, we state an alternative, equivalent, formula for the transformation of the curl + (e. g. used by Demkovicz in [12]) +

    +

    PROPOSITION 4   +For a vector field +$ \underline v$ on the tetrahedron +$ K = F_K(\hat{K})$, defined by the Piola transformation (10) of a reference + field +$ \hat{\underline v}$ on $ \hat{K}$, we have +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$ +(19)
    +

    + +

    +PROOF. + The transformation formula (19) can be proven componentwise, and + we will only carry out the proof for the first vector component +$ (\mathop{\rm curl}\underline v)_1$. The proofs for the other components + follow analogously. +
    +The identity (19) reads for the first vector component +

    +
    + + + + +
    $\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$ +(20)
    +

    + Referring to (17), we show that the right hand side of (20) equals + +$ \det \mathrm{M_1}$. + For this, we expand +$ \det \mathrm{M_1}$ to + +

    +
    +$\displaystyle \det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underlin... +...ehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,, +$ +

    + where +$ \mathcal{B}^{inv}_{ij}$ is the +$ 2 \times 2$-matrix arising from $ B_K^{-1}$ when cancelling its i-th row and its j-th column. +
    +We recall the formula for the inverse of a matrix +$ A \in \mathbb{R}^{3\times 3}$ +

    +
    + + + + +
    $\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$ +(21)
    +

    + where +$ \mathcal{A}_{ij}$ is the +$ 2 \times 2$-matrix arising from $ A$ when cancelling its i-th row and its j-th column. +
    +Replacing $ B_K$ in the right hand side of (19) by the expression (21) + for +$ A = B_K^{-1}$, we get +     +

    +
    +$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \math... +...l}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,. +$ +

    + + +

    + +

    +
    +2.2 Nédélec Elements on affine quadrilateral or hexahedral grids +

    + We want to present the ingredients for Nédélec's finite elements of first type on + grids consisiting of parallelograms (in 2d) or the respective objects in 3d, so called parallelotops (cf. section + FE built on cubes in [8]). Such grids consist + of elements $ C$ that are affine images of the square or cubic reference element +$ \hat{C} = [0,1]^d$: + +

    +
    +$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,. +$ +

    +

    + +

    +2.2.1 Polynomial spaces on the reference element +

    +

    +In order to introduce the function spaces needed for the construction of Nédélec's finite elements, + let us define some spaces of vector-valued polynomials +

    +

    DEFINITION 7   + +$ \mathcal{Q}_{l,m}$ are the spaces of polynomials on the reference square $ \hat{C}$ with maximal degree $ l$ in $ \hat{x}_1$ and + $ m$ in $ \hat{x}_2$. +
    +$ \mathcal{Q}_{l,m,n}$ are the spaces of polynomials on the reference cube $ \hat{C}$ with maximal degree $ l$ + in $ \hat{x}_1$, $ m$ in $ \hat{x}_2$ and $ n$ in $ \hat{x}_3$. +

    + +

    +The spaces $ \hat{R}$ for the reference shape functions now are in 2d +

    +
    + + + + +
    $\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{... +...t{u}_1 \in \mathcal{Q}_{k-1,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$ +(22)
    +

    + +

    +and in 3d +

    +
    + + + + +
    $\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{... +...}_2 \in \mathcal{Q}_{k,k-1,k}\,, \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$ +(23)
    +

    + +

    +We renounce an example, since it is quite evident, what these spaces look like for a specific $ k$. + +

    + +

    + +
    +2.2.2 Degrees of freedom on the reference element +

    + We start with the degrees of freedoms on the reference square +$ \hat{C}\subset \mathbb{R}^2$: +

    +

    DEFINITION 8   +Let $ \hat{C}$ denote the reference square and +$ \hat{\underline t}$ the tangent as defined in convention 1. + The set of degrees of freedom +$ \mathcal{A}$ on +$ \mathcal{P}^k$ in the 2d case consists of the linear functionals +
    +
    edge dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli... +... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$ +   
    +

    +for every edge $ \hat{e}$ of $ \hat{C}$. We have a total of $ 4k$ of edge dofs. + +

    +

    +
    inner dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin... +... }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$ +   
    +

    +We have a total of $ 2k(k-1)$ of inner dofs. + +

    +

    +

    + +

    +

    +

    DEFINITION 9   +Let $ \hat{C}$ denote the reference cube, +$ \hat{\underline t}$ the tangent to an edge as defined in convention 1 + and +$ \hat{\underline n}$ the outward unit normal vector to a face. + The set of degrees of freedom +$ \mathcal{A}$ on +$ \mathcal{P}^k$ in the 3d case consists of the linear functionals +
    +
    edge dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli... +... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$ +   
    +

    +for every edge $ \hat{e}$ of $ \hat{C}$. We have a total of $ 12k$ of edge dofs. + +

    +

    +
    face dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli... +..._{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$ +   
    +

    +for every face $ \hat{f}$ of $ \hat{C}$. We have a total of +$ 6\cdot 2k(k-1)$ of face dofs. + +

    +

    +
    inner dofs
    +

    +
    + + + + +
    $\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin... +...athcal{Q}_{k-2,k-1,k-2}\,, \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$ +   
    +

    +We have a total of $ 3k(k-1)^2$ of inner dofs. + +
    +

    + +

    +

    +

    EXAMPLE 4   +Proceeding the same way as in example 3 for a triangular reference element, we obtain +the reference shape functions of lowest +order on the square $ [0,1]^2$. For the unit tangents as in convention 1 + +

    +
    +$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array... +... +\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,, +$ +

    +they read +

    +
    + + + + +
    $\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ 0 \e... +...\underline N}_3 = \left(\begin{array}{cc} 0 \\ \hat{x}-1 \end{array}\right)\,.$ +(24)
    +

    + +

    + +

    +2.2.3 Transformation of the vector field +

    +Since the elements of the considered grids are still affine images of the reference element, we can use the Piola transformation +(10) to transform vector fields and the results stated in sections +2.1.3 - 2.1.5 can be carried over one to one. + +

    +2.3 Construction of Nédélec elements on bi- or trilinear elements +

    + We now want to consider grids that are composed of elements that are a bi- resp. trilinear images +$ F_C(\hat{C})$ of the reference + element +$ \hat{C} = [0,1]^d$. + The main difference here is, that the jacobian +$ \hat{D}F_C(\hat{x})$ of the element map $ F_C$ is not + constant, and we have to use Piola transformation (9) to + transform vector fields. +
     +

    + +

    +2.3.1 Bilinear elements in 2d +

    + The polynomial spaces +$ \mathcal{P}^k$ and the dofs remain the same as in the case of affine quadrilateral elements. +
    +A transformed vector field on a general element is now defined by the Piola transformation (9) +     +

    +
    +$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$ +

    of a vector field on the reference element. Note that + the jacobian +$ \hat{D}F_C(\hat{x})$ is not constant in this case. In contrast to the case of affine elements, the gradient +$ D\underline v$ does not transform according to + formula (12). Non-vanishing second derivatives of +$ \hat{D}F_C(\hat{x})$ appear in the transformation rule + for gradients of vector fields. + This requires a new approach to express +$ \mathop{\rm curl}\underline v$ in terms of + +$ \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. + +

    +

    +

    PROPOSITION 5   +Let $ \hat{C}$ be the reference element $ [0,1]^2$ and $ C$ a bilinear image of $ \hat{C}$. If the vector field +$ \underline v(x)$ + transforms according to the Piola transformation (9), then the transformation of the curl obeys +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,$ +   
    +

    +as in the affine case. +

    + +

    +PROOF. + In this proof, the mapped element $ C$ will be fixed, so for simplicity we write $ F$ for $ F_C$. +
    +First note that +$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$. We use the notation +$ 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 +     +

    +
    +$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,. +$ +

    + In the case of affine elements, i. e. for constant jacobian, we have + +

    +

    +
    + + + + +
    \begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part... +... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath} +   
    +

    +

    +whereas for non-constant jacobian we have + +

    +

    +
    + + + + +
    \begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part... +... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath} +   
    +

    +

    +We see that in both cases we have + +

    +
    +$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1}... +...ial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \, +$ +

    + 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. + + +

    + +

    +2.3.2 Trilinear elements in 3d +

    + The polynomial spaces +$ \mathcal{P}^k$ and the dofs remain the same as in the case of affine hexahedral elements. +
    +The + vector field on a genereal element is defined by the Piola transformation (9). +
    +The problem of the non-vanishing second derivatives of the jacobian +$ D(F_C^{-1})(x)$ arises again, and we cannot generalize the results + from the affine case straight away. +
    +But analogously to the 2d case, one can check that in in the transformation rule for expressions + +$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$, +$ i,j = 1,2,3$, + which define the curl-operator, the terms containing second derivatives vanish. We have therefore again the transformation rule + (16) for the skew matrix +$ \mathop{\rm Curl}v = Dv^T - Dv$: +     +

    +
    +$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm C... +...x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,. +$ +

    +

    +It follows that the following proposition can be proved analogously to the case of affine elements (replace there $ B_C$ by + +$ \hat{D}F_C(\hat(x))$). +

    +

    PROPOSITION 6   +Let the vector field +$ \underline v(x)$ on a trilinear image +$ C = F_C(\hat{C})$ be defined by the Piola transformation of a + reference field +$ \hat{\underline v}(\hat{x})$ on $ \hat{C}$. The transformation formula for the curl then reads +

    +
    + + + + +
    $\displaystyle \mathop{\rm curl}\underline v = \left(\frac{1}{\det \hat{D}F_C}\,... +...F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$ +   
    +

    + +

    + +

    +2.4 Construction of global shape functions +

    + 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 +$ H(\mathop {\rm curl};\Omega )$-conforming finite element method. + Indeed, in [8], Nédélec shows the invariance of the spaces + +$ \mathcal{R}^k$ and +$ \mathcal{Q}^k$ under Piola transformation of the vector field, as well as the unisolvence of the set of + degrees of freedom +$ \mathcal{A}$ from sections 2.1.2 and 2.2.2 + (for details, see [8], Section 1.2, Theorem 1 and Section 2, Theorem 5). + This leads to the fact that +$ H(\mathop {\rm curl};\Omega )$-conforming global shape functions can be defined by mapping elementwise the reference + shape functions with the Piola transformation +$ \mathcal{P}_K$. + However, we must pay some care to the orientation of an interface on which the moments defining the degrees of freedom are based. + For the 2d case, we will illustrate in this section how we must take into account the orientation of an edge + in the definition of the respective element edge shape function, in order to get an +$ H(\mathop {\rm curl};\Omega )$-conforming finite element space + of global shape functions. + +
     +

    +Let +$ K=F(\hat{K})$ be an affine or bilinear image of a reference element, $ e$ one of its edges and $ \hat{e}$ the + corresponding edge on the reference element. +
    +Let further +$ [0,\vert e\vert] \ni s \mapsto \underline x(s) \in e$ and +$ [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 $ e$ and $ \hat{e}$ respectively. We can assume that these parametrizations endow the edges with a + counterclockwise orientation. Then, the unit tangent vectors +$ \underline t$ and +$ \hat{\underline t}$ are given by + +$ \frac{d \underline x}{ds}$ and +$ \frac{d \hat{\underline x}}{d\hat{s}}$. +     +

    +

    +

    LEMMA 1   +Let +$ \hat{\underline v}(\hat{x})$ be a vector field on the reference element and +$ \underline v(x)$ be the corresponding vector field on $ K$, defined by the + Piola transformation (9). It then holds +

    +
    + + + + +
    $\displaystyle \underline v\cdot \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert} (\hat{\underline v}\cdot \hat{\underline t})\,,$ +(25)
    +

    +where $ \vert\hat{e}\vert$ and $ \vert e\vert$ denote the length of the edges $ \hat{e}$ and $ e$. +

    + +

    +PROOF. + With + +

    +
    +$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\, +$ +

    + and +$ \hat{x}_j = \hat{x}_j(\underline x(s))$ and +$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$ on the edges, we have + +

    +
    +$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \under... +...t{s}}{ds} += (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds} +$ +

    + and with +$ \frac{d\hat{s}}{ds}=\frac{\vert\hat{e}\vert}{\vert e\vert}$ the lemma follows. + + +

    +As a consequence, we have + +

    +

    +

    PROPOSITION 7 (Invariance of the edge dofs)   +Let the vector field +$ \underline v(x)$ on $ K$ be defined by the Piola transformation (9) of a reference vector field + +$ \hat{\underline v}(\hat{x})$ on $ \hat{K}$. Then, the functionals (edge dofs) + +$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$ are invariant in the sense of + +

    +
    +$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline... +...{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,. +$ +

    + +

    +Let now +$ K_- = F_-(\hat{K})$ and +$ K_+ = F_+(\hat{K})$ be two neighbouring triangles with common edge $ e$. Let +$ \underline N$ be the global edge shape + function that 'lives' on $ e$. By +$ \underline N_-$ and +$ \underline N_+$ we denote the restriction of +$ \underline N$ to $ K_+$ and $ K_-$ respectively. + Let +$ e_+ =F_+(\hat{e}_i)$ and +$ e_- =F_-(\hat{e}_j)$. + We write +$ \underline t_+ $ for the tangential unit vector to $ e$, oriented + counterclockwise with respect to $ K_+$ and +$ \underline t_- = -\underline t_+$ + for the respective from $ K_-$. For line integrals over the edge $ e$ we write +$ \int_{e_+}$ if we chose the orientation induced by + +$ \underline t_+ $ and +$ \int_{e_-}$ for the orientation of $ e$ induced by +$ \underline t_-$. +
    +In order to obtain an +$ H(\mathop {\rm curl};\Omega )$-conforming method, proposition 1 tells us that we must ensure the continuity of the + tangential components of the global shape functions, that is +

    +
    + + + + +
    $\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$ +(26)
    +

    + 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 +$ \alpha^{[K_+]}$ and +$ \alpha^{[K_-]}$ guarantees + the pointwise condition (26). +

    +

    LEMMA 2   +Let $ \hat{K}$ denote the reference triangle and $ \hat{e}$ one of its edges, parametrized by +$ \hat{e} \ni \hat{x}(s) := \underline a + +s\, \hat{\underline t}$. + Let +$ \hat{\underline p} \in \mathcal{S}^k$, +$ \mathcal{S}^k$ as defined in (4). It then holds + +

    +
    +$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,. +$ +

    + +

    +PROOF. + +

    +
    +$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad ... +...{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad +\textrm{where } \sum_{j=1}^3 k_{ij} = k\,. +$ +

    + Hence, with the parametrization of $ \hat{e}$ by +$ \hat{x}(s)$ + +

    +
    +$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} + +\hat{\varphi }_{k-1}(s)\,, +$ +

    + with +$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$, + and + +

    +
    +$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} = s^... +...t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) + +\hat{\varphi }_{k-1}(s)\,. +$ +

    + We observe that the coefficient of $ s^k$ is exactly +$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$. By the definition of the space + +$ \mathcal{S}^k$, this expression must vanish. + + +

    +

    +

    REMARK 5   +In the case of $ \hat{K}$ being a quadrilateral, we have +$ \hat{R} = \mathcal{P}^k$. By the definition of +$ \mathcal{P}^k$ we see + immediately that here also +$ (\hat{\underline v}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$. +

    + +

    +The next proposition tells us how exactly to define element shape functions on a mapped element $ K$ in order to get + +$ H(\mathop {\rm curl};\Omega )$-conforming global shape functions. +

    +

    PROPOSITION 8   +Condition (26) is satiesfied, if we define the element shape functions +$ \underline N_+$ and +$ \underline N_-$ + by the Piola transformation (10) and take into account the orientation of the edge $ e$: +

    +
    + + + + +
    $\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_... +...\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$ +(27)
    +

    + +

    +PROOF. + Let $ \hat{K}$ be the reference element and $ K$ its affine or bilinear image. + Let +$ \underline v := \mathcal{P}_K(\hat{\underline v})$ be a vector field on $ K$, defined by the Piola transformation of a reference vector field + +$ \hat{\underline v} \in \hat{R}$. Let $ e$ be one of the edges of $ K$ and +$ \underline t$ + the tangent according to convention 1. +
    +In the case of $ \hat{K}$ being a triangle, we have +$ \hat{R} = \mathcal{R}^k$. + By the definition of the space +$ \mathcal{R}^k$, lemma 2 and 1 we can + conclude that +$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$. +
    +If $ \hat{K}$ is a quadrilateral, the previous remark and + 1 also tell us that +$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$. +
    +Hence the condition +     +

    +
    +$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underli... +...ne t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e) +$ +

    + on the edge moments + is sufficient for the global edge shape functions to satiesfy (26). + Note that +
    +$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$. + So, by the definition (27) of the element shape functions on $ K_+$ resp. on $ K_-$, + by the invariance of the dofs (proposition 7) and by the definition of the + reference shape functions (example 3) we have +

    +
    + + + + +
    $\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \i... +...\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$ +   
    +

    + + +

    +To close this section, let us make a note on the interpretation of the dofs on an element $ K$ in the case of lowest order polynomial + degree. In this case, all dofs are edge dofs, the degrees of freedom are +$ \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 +$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$ for the reference shape functions, we have + +

    +
    +$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j ... +...\hat{e}_j\vert = ({\underline N}_j \cdot {\underline t}_j)\, \vert e_j\vert\,, +$ +

    + where for the last equality we have used lemma 1. We see that + the dof +$ \alpha_j(\underline v)$ 'sitting' on the edge $ e_j$ is the value of the scaled tangential component + +$ \vert e_j\vert\left(\underline v\cdot \underline t_j\right)\vert _e$. +

    +

    REMARK 6   +For the invariance of the edge dofs it is essential that the moments +$ \alpha^{[K]}$ on $ K$ are defined by using the + unit tangent + vector +$ \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert}\,(\hat{D}F)\,\hat{\underline t}$ on $ K$. If not, e. g. if we + just used the tangent +$ \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 ([8], remark on p. 326). +

    + +

    + +

    +2.5 Approximation and convergence results +

    + Without going into details, we will cite here some results on approximation properties and convergence of Nédélec FEM of first type. + +
    +We are in the setting of a + conforming FEM and have quasi-optimal approximation properties of the FE-spaces +$ V_h \subset H(\mathop{\rm curl};\Omega )$ +     +

    +
    +$\displaystyle \Vert \underline u - \Pi_h^k \underline u\Vert _{H(\mathop{\rm cu... +...V_h}\Vert \underline u - \underline w\Vert _{H(\mathop{\rm curl}; \Omega )}\,, +$ +

    + where +$ \Pi_h^k \underline u \in \mathcal{R}^k$ or +$ \Pi_h^k \underline u \in \mathcal{P}^k$ respectively, + denotes the interpolate of +$ \underline u$ with regard to the Nédélec dofs: +$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$ for all dofs $ \alpha$. + The interpolation operator $ \Pi_h^k$ is defined for sufficiently smooth vector fields, namely for all +$ \underline v\in H^r(\mathop{\rm curl})$ + for any +$ r>\frac{1}{2}$ (see [1], Lemma 5.1., [7] and references therein). +
    +For Nédélec's FEM of first type we state (without proof) the following optimal estimate in the curl-norm: +

    +

    THEOREM 5   +If +$ \mathcal{T}_h$, $ h>0$, is a regular family of triangulations on $ \Omega $ and +$ r>\frac{1}{2}$, then there exists a constant $ C>0$, + depending on $ r$ but not on $ h$ or +$ \underline v$, such that +

    +
    + + + + +
    $\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{H(\mathop{\rm cu... +... C\,h^{\min\{r,k\}} \Vert\underline v\Vert _{H^r(\mathop{\rm curl};\Omega )}\,,$ +(28)
    +

    +for all +$ \underline v\in H^r(\mathop{\rm curl};\Omega )$. +

    + + The result in (28) was obtained by Alonso and Valli in [1], extending earlier interpolation + results by Nédélec in [8] and Monk in [6]. +
    +Optimal convergence in the +$ H(\mathop {\rm curl};\Omega )$-norm for the error of the FE-approximation of the model + problem (3) by Nédélec's elements of first type follows from (28) by Céa's lemma. + This result has been verified in numerical experiments with a MATLAB code, which uses lowest + order Nédélec elements on affine triangular meshes for 2d problems, as well as with a deal.II code, which uses lowest order + Nédélec elements on bilinear resp. trilinear meshes for 2d resp. 3d problems. +
    +As for the +$ L^2(\Omega )$-approximation properties of FE spaces based on +$ \mathcal{R}^k$ or +$ \mathcal{P}^k$, we could hope for a better order + than +$ \mathcal{O}(h^k)$ at first sight: still, we have +$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$. However, Nédélec shows in [8] + that only suboptimality can be expected: +

    +
    + + + + +
    $\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{L^2(\Omega )} \leq C h^k \vert\underline v\vert _{H^k(\Omega )}\,.$ +(29)
    +

    + Nédélec uses a standard scaling and Bramble-Hilbert argument to derive (29). Since + +$ [\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 $ k$ of $ H^k(K)$-functions from the space +$ \mathcal{R}^k(K)$. +
    +However, in a recent paper Hiptmair uses a duality technique to state optimal convergence of the + +$ L^2(\Omega )$-error +$ \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )}$ + for the 3d case and Nédélec's elements of first type of order $ k$ on tetrahedral meshes + (see Section 5.3, Theorem 5.8 in [5]): +

    +

    THEOREM 6   +There is an +$ s>\frac{1}{2}$ such that +

    +
    + + + + +
    $\displaystyle \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )} \leq C h^s \Vert \underline u - \underline u_h\Vert _{H(\mathop{\rm curl}; \Omega )}\,.$ +(30)
    +

    +Under the assumption that the boundary +$ \partial \Omega $ is smooth or convex, $ s=1$ can be chosen. +

    + + Several key arguments of the proof in [5] 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 (30). +
    +A possibility to overcome this deficiency of convergence is to use Nédélec elements of second type, where the full +$ [\mathbb{P}_k]^d$ + are used as polynomial spaces (see [10]). +     +

    + +

    +
    + +2003-04-30 +
    + + diff --git a/deal.II/doc/reports/nedelec/node3.html b/deal.II/doc/reports/nedelec/node3.html new file mode 100644 index 0000000000..221339e7cd --- /dev/null +++ b/deal.II/doc/reports/nedelec/node3.html @@ -0,0 +1,720 @@ + + + + + +3 Numerical results + + + + + + + + + + + + + + + +

    +3 Numerical results +

    + The numerical results in this section provide some samples of the quality of the +$ H(\mathop {\rm curl};\Omega )$-conforming FEM + with Nédélec elements of first type and lowest order (polynomial degree $ k=1$). +
    +We considered the model problem (1) in +$ \Omega = [-1,1]^d$, $ d=2,3$, + with homogeneous Dirichlet boundary condition (2). +
    +The first few results for the two-dimensional problem have been obtained by a MATLAB code. + For the first example we used the data +

    +
    + + + + +
    $\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\ 3 - x^2 \end{array}\right)\,.$ +(31)
    +

    + For the second example we have followed the outlines from Appendix A and taken the data from example 5 +

    +
    + + + + +
    $\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,.$ +(32)
    +

    + The finite element solution has been computed using Nédélec elements of first type and of polynomial degree $ k=1$ on a family of + affine triangular grids. + The initial coarse grid consisted of $ 2^5$ triangles. The finest grid with $ 2^{13}$ triangles results after five global refinements. +
    +In Table 1 we see that for both examples we have + optimal convergence in the +$ H(\mathop {\rm curl};\Omega )$-semiorm, as we would expect from the theoretical results of the + previous section. As for the +$ 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 Nédélec elements of first + type and of polynomial degree $ k=1$, we got only +$ \mathcal{O}(h)$-convergence of the $ L^2$-error. However, this order of + convergence is consistent with the result (29) obtained by Nédélec in [8]. +
     +

    +

    +
    + + + +
    Table 1: +Errors and convergence rates in the +$ L^2(\Omega )$-norm and +$ H(\mathop{\rm curl};\Omega )$-seminorm for the two MATLAB examples.
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
     grid$ \char93 $ cells +$ H(\mathop{\rm curl})$-error$ L^2$-error
     1326.66e-01-4.66e-01-
     21283.33e-011.002.35e-010.98
    example 135121.66e-011.001.17e-010.99
     420488.33e-021.005.89e-020.99
     581924.17e-021.002.95e-020.99
     1323.05e+00-6.48e-01-
     21281.61e+000.913.22e-011.00
    example 235120.81e-010.971.60e-011.00
     420480.41e-010.998.02e-021.00
     581922.05e-010.994.01e-021.00
    +
    +
    +

    +
    +

    + +

    +
    + + + +
    Figure 1: +Convergence of the FE-approximation to the smooth solution of the MATLAB example (31) + in the +$ L^2(\Omega )$-norm and the +$ H(\mathop{\rm curl};(\Omega ))$-seminorm
    +
    + +% latex2html id marker 4265 +\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps} + +
    +

    + +

    + +

    +
    + + + +
    Figure 2: +Convergence of the FE-approximation to the smooth solution of the MATLAB example (32) + in the +$ L^2(\Omega )$-norm and the +$ H(\mathop{\rm curl};(\Omega ))$-seminorm
    +
    + +% latex2html id marker 4271 +\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps} + +
    +

    + +

    +

    +

    REMARK 7   +The mesh generation and refinement was done by PDE-toolbox commands. Since the PDE-toolbox 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. +

    + +

    +As for meshes with quadrilateral cells, numerical results were obtained with a deal.II code, + using the finite element class fe/fenedelec.cc. This class provides + Nédélec's +$ 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 deal.II, see [2]. + In the following results were obtained for the model problem (1) in two dimensions using the + data (32). + We computed the solution on five successive + non-affine bilinear grids ( figure 3), each of which was obtained by global refinement of the previous one. + +

    + +

    +
    + + + +
    Figure 3: +Non-affine bilinear grid used in the deal.II code, after one refinement step.
    +
    + +\includegraphics[width=5.5cm, height=5.5cm]{grid.eps} + +
    +

    + +

    +Again, in Table 2 we can observe optimal convergence of order +$ \mathcal{O}(h)$ in the +$ H(\mathop {\rm curl};\Omega )$-norm. +The same order of convergence is obtained for the error in the +$ L^2(\Omega )$-norm. +
     +

    +

    +
    + + + +
    Table 2: +Errors and convergence rates in the +$ H(\mathop{\rm curl};\Omega )$- and +$ L^2(\Omega )$-norm for the 2d-example solved with deal.II. +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    grid$ \char93 $ cells +$ H(\mathop{\rm curl})$-error$ L^2$-error
    146.112e+00-1.442e+00-
    2163.688e+000.736.765e-011.09
    3641.991e+000.893.280e-011.04
    42561.015e+000.971.617e-011.02
    510245.098e-010.998.049e-021.01
    +
    +
    +

    +
    +

    +With deal.II, we are also able to treat 3d problems on hexahedral grids. For our type of problem, +Nédélec's +$ H(\mathop {\rm curl};\Omega )$-conforming elements of first type and lowest order, based on a cubic reference element, are available. +
    +We computed an approximation to the model problem (1) in 3d using the data +

    +
    + + + + +
    $\displaystyle c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{cc... +... + (1-y^2)(2-x^2-z^2) \\ yz(1 - x^2)(1-y^2) + 2yz(1-x^2) \end{array}\right)\,.$ +(33)
    +

    +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. +
    +We see in Table 3 that in both cases we observe again convergence of order +$ \mathcal{O}(h)$ in the + +$ H(\mathop {\rm curl};\Omega )$- and the +$ L^2(\Omega )$-norm. +
     +

    +

    +
    + + + +
    Table 3: +Errors and convergence rates in the +$ H(\mathop{\rm curl};\Omega )$- and +$ L^2(\Omega )$-norm for the 3d-example solved with deal.II. The first data +set is for the computation on a family of affine grids, the second set of data is for non-affine trilinear grids.
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
     grid$ \char93 $ cells +$ H(\mathop{\rm curl})$-error$ L^2$-error
     187.696e-01-6.609e-01-
     2644.088e-010.912.943e-011.17
    affine grids35122.075e-010.981.408e-011.06
     440961.041e-010.996.955e-021.02
     5327685.210e-021.003.467e-021.00
     187.716e-01-6.611e-01-
     2644.108e-010.912.955e-011.16
    non-affine grids35122.085e-010.981.413e-011.06
     440961.046e-010.996.982e-021.02
     5327685.237e-021.003.480e-021.00
    +
    +
    +

    +
    +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 $ L^2$-convergence of the error (6) cannot be relaxed. +
     +

    +Finally, here are some pretty pictures: the vector field plots from the MATLAB computations. + +

    + +

    +
    + + + +
    Figure 4: +Vector-field plot of the FE-solution of example (31).
    +
    + +\includegraphics[width=9.5cm, height=7cm]{field1.eps} + +
    +

    + +

    + +

    +
    + + + +
    Figure 5: +Vector-field plot of the FE-solution of example (32).
    +
    + +\includegraphics[width=9.5cm, height=7cm]{field2.eps} + +
    +

    + + +

    + +

    +

    +
    + +2003-04-30 +
    + + 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 @@ + + + + + +Bibliography + + + + + + + + + + + + + + + +

    +Bibliography +

    1 +
    +A. Alonso and A. Valli. +
    An optimal domain decomposition preconditioner for low-frequency + time-harmonic Maxwell equations. +
    Math. Comp., 68(226):607-631, 1999. + +

    2 +
    +W. Bangerth, R. Hartmann, and G. Kanschat. +
    deal.II Differential Equations Analysis Library, + Technical Reference. +
    IWR, Universität Heidelberg. +
    http://www.dealii.org. + +

    3 +
    +F. Brezzi and M. Fortin. +
    Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. +
    Springer-Verlag, New York, 1991. + +

    4 +
    +V. Girault and P.-A. Raviart. +
    Finite Element Approximation of the Navier-Stokes Equations, + volume 749 of Lecture Notes in Mathematics. +
    Springer-Verlag, Berlin, Heidelberg, 1979, 1981. + +

    5 +
    +R. Hiptmair. +
    Finite elements in computational electromagnetism. +
    In Acta Numerica, pages 1-103. Cambridge University press, + 2002. + +

    6 +
    +P. Monk. +
    Analysis of a finite element method for Maxwell's equations. +
    SIAM J. Numer. Anal, 29:714-729, 1992. + +

    7 +
    +P. Monk. +
    A simple proof for an edge element discretization of Maxwell's + equations. +
    Submitted for publication. Download version available on Monk's + webpage: www.math.udel.edu./ monk, 2001. + +

    8 +
    +J. C. Nédélec. +
    Mixed finite elements in +$ \mathbb{R}^3$. +
    Numer. Math., 35:315-341, 1980. + +

    9 +
    +J. C. Nédélec. +
    Elements finis mixtes incompressibles pour l'équation de Stokes + dans +$ \mathbb{R}^3$. +
    Numer. Math., 39:97-112, 1982. + +

    10 +
    +J. C. Nédélec. +
    A new family of mixed finite elements in +$ \mathbb{R}^3$. +
    Numer. Math., 50:57-81, 1986. + +

    11 +
    +W. Rachowicz and L. Demkowicz. +
    A two-dimensional hp-adaptive finite element package for + electromagnetics (2Dhp90_EM). +
    Ticam Report 98-16, TICAM, 1998. +
    Download version available on Demkowicz' webpage: + www.ticam.utexas.edu/ Leszek. + +

    12 +
    +W. Rachowicz and L. Demkowicz. +
    A three-dimensional hp-adaptive finite element package for + electromagnetics (3Dhp90_EM). +
    Ticam Report 00-04.2000, TICAM, 2000. +
    Download version available on Demkowicz' webpage: + www.ticam.utexas.edu/ Leszek. +
    + +
    +
    + +2003-04-30 +
    + + 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 @@ + + + + + +A. Construction of solutions in 2d + + + + + + + + + + + + + + + +

    +A. Construction of solutions in 2d +

    +

    +We present how divergence-free solutions of the model problem (1) on a domain +$ \Omega \subset\mathbb{R}^2$ with perfectly + conducting boundary + can be constructed from solutions of the scalar Laplace equation. +
     +

    +

    +

    PROPOSITION 9   +Let $ \Omega $ be a sufficiently smooth domain in +$ \mathbb{R}^2$, +$ \varphi (x,y)$ a sufficently smooth scalar function on $ \Omega $ and the coefficient $ c>0$ + globally constant. +
    +Let $ w$ be a solution of the scalar equation +

    +
    + + + + +
    \begin{displaymath}\begin{split}-\Delta w + c\, w &= \varphi \quad \mathrm{in} \... +... w &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath} +(34)
    +

    +Then, +$ \underline E := \nabla^{\perp} w$ is a solution of the model equation +

    +
    + + + + +
    \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} +   
    +

    +with right hand side +$ \underline f := \nabla^{\perp} \varphi $. +
    +We use the notation +$ \nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\ -\partial _x\varphi +\end{array}\right)$. +

    + +

    +PROOF. + We first show the correspondence of the boundary conditions. With the definition +$ \underline E := \nabla^{\perp} w$ it holds + +

    +
    +$\displaystyle \underline E \wedge \underline n = \underline E \cdot \underline ... +...boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,. +$ +

    + It remains to show that +$ \underline E$ solves the model problem for an appropriate right hand side. + First, note that +$ \underline E$ is divergence-free: + +$ \nabla\cdot\nabla^{\perp}w = 0$ for all $ w$. Hence, the identity +$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$ + reduces to +$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$. The observation that for smooth data +$ \nabla^{\perp}w$ solves the Laplace equation + (34) with right hand side +$ \nabla^{\perp} \varphi $ concludes the proof. + + +

    +

    +

    EXAMPLE 5 (Solutions from eigenfunctions of the Laplacian)   +Choose $ w$ to be a solution of the eigenvalue problem +

    +
    + + + + +
    \begin{displaymath}\begin{split}-\Delta w &= \lambda \, w \quad \mathrm{in} \qua... +... w &= 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath} +   
    +

    +and set +$ \varphi = (\lambda + c)\,w$. +
    +As an example, take +$ \Omega = [-1,1]^2$ and +$ \lambda = 2\pi^2$. Then, +$ w = \cos\pi x\cos\pi y$ is an eigenfunction and we compute + +

    +
    +$\displaystyle \underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\... +...in{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,. +$ +

    + +

    +

    +

    EXAMPLE 6 (Solutions from any scalar function satiesfying the boundary condition)   +Take again +$ \Omega = [-1,1]^2$. We have to find a scalar function $ w$ which satiesfies the homogeneous Neumann boundary condition. Take + for example +$ w(x,y) = (1-x^2)^2(1-y^2)^2$, for which we have +$ \underline n \cdot \nabla w = 0$ on +$ \partial [-1,1]^2$. The right hand side is then + +$ \varphi = -\Delta w + c w$. +

    + +

    + + +

    +
    + +2003-04-30 +
    + + 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 @@ + + + + + +A. Time-harmonic Maxwell's equations with low-frequency approximation + + + + + + + + + + + + + + + +Subsections + + + +
    + +

    + +
    +A. Time-harmonic Maxwell's equations with low-frequency approximation +

    +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 [1]: +
    +We consider the following primal formulation of Maxwell's equations: +

    +
    + + + + +
    \begin{displaymath}\begin{split}\varepsilon \frac{\partial \mathcal{E}}{\partial... +...}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,, \end{split}\end{displaymath} +(35)
    +

    +where +$ \mathcal{E}$ and +$ \mathcal{H}$ are the electric and magnetic field. + +$ \varepsilon (x), \mu(x)$ are the dielectric and magnetic permeability coefficients, and $ \sigma(x)$ denotes the electric conductivity. + +$ \varepsilon (x), \mu(x)$ and $ \sigma(x)$ are assumed to be symmetric matrices in +$ L^{\infty}(\Omega )^{d\times d}$, and +$ \varepsilon (x)$ and $ \mu(x)$ are +positive definite. $ \sigma(x)$ is positive definite in a conductor and vanishes in an insulator. + +

    + +

    +Time-harmonic, low-frequency case +

    +We assume that +$ \mathcal{E}(x,t)$ and +$ \mathcal{H}(x,t)$ are time-harmonic, i. e. they can be represented as +

    +
    + + + + +
    \begin{displaymath}\begin{split}\mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(... +...= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,. \end{split}\end{displaymath} +   
    +

    +Here, +$ E(x), H(x)$ are complex-valued vector fields and +$ \omega\neq 0$ is a given angular frequency. +

    +

    REMARK 8   +For example, a monofrequent laser can be described by the time-harmonic Maxwell's equations.

    + +

    +In the time-harmonic case the space and time variables decouple and we can eliminate the time dependency. For this, we ask + +$ E(x) \exp(i\omega t)$ and +$ H(x) \exp(i\omega t)$ to satiesfy (35). +By then inserting the second equation of (35) into the first one, we can eliminate the magnetic field $ H(x)$. This yields + +

    +
    +$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0 +$ +

    +In the low-frequency case where $ \vert\omega\vert$ is small, it is known that for general materials the material parameters are such that + +

    +
    +$\displaystyle \omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,. +$ +

    +Hence, neglecting the expression +$ \omega^2\varepsilon E(x)$ is reasonable and it brings us to the low-frequency approximation of the +time-harmonic Maxwell's equations: + +

    +
    +$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0 +$ +

    +

    +We consider this equation in a conductor $ \Omega $ ($ \sigma(x)$ pos. def.) and a impose Dirichlet boundary condition on the tangential trace +of the field: +

    +
    + + + + +
    $\displaystyle E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.$ +(36)
    +

    +Proceeding as in [1], we assume that a vector funciton $ \tilde{E}$ is known, +satiesfying (36), and we end up with the following boundary value problem for +$ \underline u = E - \tilde{E}$ +

    +
    + + + + +
    \begin{displaymath}\begin{split}\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) +... +... n &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath} +(37)
    +

    +Although problem (37) is complex-valued, finding a finite element method to approximate (37) +basically boils down to finding a finite element method for the real valued model problem (1). + +

    +

    +
    + +2003-04-30 +
    + + 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 @@   MinRes linear solver.
  • Anna Schneebeli: - Help and advice for Nedelec elements. + Help and advice for Nedelec elements, writing the excellent + report on Nedelec elements.
  • Jan Schrage: Initial parts of the tutorial. -- 2.39.5