--- /dev/null
+\batchmode
+
+
+\documentclass[a4paper,11pt]{article}
+\RequirePackage{ifthen}
+
+
+\NeedsTeXFormat{LaTeX2e}
+\usepackage{exscale}
+\usepackage[dvips]{graphicx}
+\usepackage{shortvrb}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{graphicx}
+
+\addtolength{\topmargin}{-35pt}\addtolength{\headsep}{-2pt}\addtolength{\topskip}{-5pt}\addtolength{\oddsidemargin}{-1.5cm}\addtolength{\evensidemargin}{-1.5cm}\addtolength{\textheight}{45pt}%% less white space at bottom of page
+
+\addtolength{\textwidth}{4cm}%% larger columns
+
+
+%
+\providecommand{\vect}[1]{\underline{#1}}%% vectors%
+\providecommand{\matr}[1]{\mathbf{#1}}%% matrices%
+\providecommand{\ofx}{(\underline{x})}%
+\providecommand{\oftx}{(t,\underline{x})}%
+\providecommand{\R}{\mathbb{R}}%% number sets%
+\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}
+
+\usepackage[latin1]{inputenc}
+
+
+
+\makeatletter
+
+\makeatletter
+\count@=\the\catcode`\_ \catcode`\_=8
+\newenvironment{tex2html_wrap}{}{}%
+\catcode`\<=12\catcode`\_=\count@
+\newcommand{\providedcommand}[1]{\expandafter\providecommand\csname #1\endcsname}%
+\newcommand{\renewedcommand}[1]{\expandafter\providecommand\csname #1\endcsname{}%
+ \expandafter\renewcommand\csname #1\endcsname}%
+\newcommand{\newedenvironment}[1]{\newenvironment{#1}{}{}\renewenvironment{#1}}%
+\let\newedcommand\renewedcommand
+\let\renewedenvironment\newedenvironment
+\makeatother
+\let\mathon=$
+\let\mathoff=$
+\ifx\AtBeginDocument\undefined \newcommand{\AtBeginDocument}[1]{}\fi
+\newbox\sizebox
+\setlength{\hoffset}{0pt}\setlength{\voffset}{0pt}
+\addtolength{\textheight}{\footskip}\setlength{\footskip}{0pt}
+\addtolength{\textheight}{\topmargin}\setlength{\topmargin}{0pt}
+\addtolength{\textheight}{\headheight}\setlength{\headheight}{0pt}
+\addtolength{\textheight}{\headsep}\setlength{\headsep}{0pt}
+\setlength{\textwidth}{349pt}
+\newwrite\lthtmlwrite
+\makeatletter
+\let\realnormalsize=\normalsize
+\global\topskip=2sp
+\def\preveqno{}\let\real@float=\@float \let\realend@float=\end@float
+\def\@float{\let\@savefreelist\@freelist\real@float}
+\def\liih@math{\ifmmode$\else\bad@math\fi}
+\def\end@float{\realend@float\global\let\@freelist\@savefreelist}
+\let\real@dbflt=\@dbflt \let\end@dblfloat=\end@float
+\let\@largefloatcheck=\relax
+\let\if@boxedmulticols=\iftrue
+\def\@dbflt{\let\@savefreelist\@freelist\real@dbflt}
+\def\adjustnormalsize{\def\normalsize{\mathsurround=0pt \realnormalsize
+ \parindent=0pt\abovedisplayskip=0pt\belowdisplayskip=0pt}%
+ \def\phantompar{\csname par\endcsname}\normalsize}%
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+\newcommand\lthtmlvboxmathA{\adjustnormalsize\setbox\sizebox=\vbox\bgroup %
+ \let\ifinner=\iffalse \let\)\liih@math }%
+\newcommand\lthtmlboxmathZ{\@next\next\@currlist{}{\def\next{\voidb@x}}%
+ \expandafter\box\next\egroup}%
+\newcommand\lthtmlmathtype[1]{\gdef\lthtmlmathenv{#1}}%
+\newcommand\lthtmllogmath{\lthtmltypeout{l2hSize %
+:\lthtmlmathenv:\the\ht\sizebox::\the\dp\sizebox::\the\wd\sizebox.\preveqno}}%
+\newcommand\lthtmlfigureA[1]{\let\@savefreelist\@freelist
+ \lthtmlmathtype{#1}\lthtmlvboxmathA}%
+\newcommand\lthtmlpictureA{\bgroup\catcode`\_=8 \lthtmlpictureB}%
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+ \vrule height1.5ex width0pt }%
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+ \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetmath}
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+ \centerinlinemath\lthtmllogmath\lthtmlsetmath}
+\def\lthtmlsetinline{\hbox{\vrule width.1em \vtop{\vbox{%
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+ \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 \,. \end{split}\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+
+\end{document}
--- /dev/null
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--- /dev/null
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 2K.1beta (1.47)
+original version by: Nikos Drakos, CBLU, University of Leeds
+* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+ Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>2 Nédélec's elements of first type for </TITLE>
+<META NAME="description" CONTENT="2 Nédélec's elements of first type for ">
+<META NAME="keywords" CONTENT="main">
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+
+</HEAD>
+
+<BODY >
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
+
+<UL>
+<LI><A NAME="tex2html59"
+ HREF="#SECTION00021000000000000000">2.1 Construction of Nédélec elements on tetrahedral grids</A>
+<UL>
+<LI><A NAME="tex2html60"
+ HREF="#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html61"
+ HREF="#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html62"
+ HREF="#SECTION00021300000000000000">2.1.3 Piola transformation</A>
+<LI><A NAME="tex2html63"
+ HREF="#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
+<LI><A NAME="tex2html64"
+ HREF="#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html65"
+ HREF="#SECTION00022000000000000000">2.2 Nédélec Elements on affine quadrilateral or hexahedral grids</A>
+<UL>
+<LI><A NAME="tex2html66"
+ HREF="#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
+<LI><A NAME="tex2html67"
+ HREF="#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
+<LI><A NAME="tex2html68"
+ HREF="#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html69"
+ HREF="#SECTION00023000000000000000">2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
+<UL>
+<LI><A NAME="tex2html70"
+ HREF="#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
+<LI><A NAME="tex2html71"
+ HREF="#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
+</UL>
+<BR>
+<LI><A NAME="tex2html72"
+ HREF="#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
+<LI><A NAME="tex2html73"
+ HREF="#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
+</UL>
+<!--End of Table of Child-Links-->
+<HR>
+
+<H1><A NAME="SECTION00020000000000000000">
+2 <FONT SIZE="+1">Nédélec's elements of first type for <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
+</H1><FONT SIZE="-1">
+ In this section we will present present <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming vector-valued finite elements,
+ the Nédélec elements of first type (cf. [<A
+ HREF="node4.html#Ned1">8</A>]), which can be used to discretize the variational problem (<A HREF="node1.html#eq:_varform">3</A>).
+<BR></FONT>
+<P>
+<FONT SIZE="-1">In order to define a finite element we must specify
+ </FONT><DL>
+<DT><STRONG>the geometry</STRONG></DT>
+<DD>We choose a reference element <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> and a change of variables <!-- MATH
+ $F_K(\hat{x})$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img82.gif"
+ ALT="$ F_K(\hat{x})$">, the element map.
+ We set <!-- MATH
+ $K = F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$">.
+
+</DD>
+<DT><STRONG>a function space</STRONG></DT>
+<DD>We need a <I>finite dimensional</I> function space <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$">, typically a space of polynomials, on the reference
+ cell, plus a transformation of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> to a function space <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img85.gif"
+ ALT="$ R_K$"> on a general cell <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">.
+
+</DD>
+<DT><STRONG>dofs</STRONG></DT>
+<DD>We have to define a set of dofs <!-- MATH
+ $\mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$
+ -->
+<IMG
+ WIDTH="119" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img87.gif"
+ ALT="$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$">. These are linear functionals on
+ <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> and <!-- MATH
+ $N < \infty$
+ -->
+<IMG
+ WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img88.gif"
+ ALT="$ N < \infty$"> is the dimension of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$">. <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> should be <I>unisolvent</I>, that is, the dofs <!-- MATH
+ $\alpha_i(\cdot)$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img90.gif"
+ ALT="$ \alpha_i(\cdot)$">
+ are linearly independent.
+
+</DD>
+</DL>
+<P>
+<FONT SIZE="-1">First, we observe that for a conforming discretization of (<A HREF="node1.html#eq:_varform">3</A>) we cannot take vector-valued finite elements
+ that are build by taking the standard nodal finite element spaces of globally continuous functions for each vector component.
+ For <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-functions, the only continuity condition is the continuity of the tangential component over cell boundaries.
+ This fact will motivate the choice of appropriate degrees of freedom (abbreviated by dofs in the following).
+<BR>
+We will give an outline of the construction of the finite element spaces described by Nédélec in [<A
+ HREF="node4.html#Ned1">8</A>]. In literature, they are also referred to as
+ <I>Nédélec's elements of first type</I>.
+<BR></FONT>
+<P>
+
+<H2><A NAME="SECTION00021000000000000000">
+2.1 Construction of Nédélec elements on tetrahedral grids</A>
+</H2>
+<P>
+<FONT SIZE="-1">In this section, we denote by <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> the standard triangular or tetrahedral reference element.
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00021100000000000000">
+2.1.1 Polynomial spaces on the reference element</A>
+</H3><FONT SIZE="-1">
+ In [<A
+ HREF="node4.html#Ned1">8</A>], Nédélec introduces the function spaces <!-- MATH
+ $\hat{R} = \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img91.gif"
+ ALT="$ \hat{R} = \mathcal{R}^k$">, on which his finite element will be based. These spaces are subject to this
+ section.
+ For more details, consult [<A
+ HREF="node4.html#Ned1">8</A>].
+<BR>
+We denote by <!-- MATH
+ $\mathbb{P}_k(\hat{\Sigma})$
+ -->
+<IMG
+ WIDTH="51" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img92.gif"
+ ALT="$ \mathbb{P}_k(\hat{\Sigma})$"> the space of polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> on <!-- MATH
+ $\hat{\Sigma}$
+ -->
+<IMG
+ WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img94.gif"
+ ALT="$ \hat{\Sigma}$">, where <!-- MATH
+ $\hat{\Sigma}$
+ -->
+<IMG
+ WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img94.gif"
+ ALT="$ \hat{\Sigma}$"> is an edge, a
+ face of or the reference element itself.. The space <!-- MATH
+ $\tilde{\mathbb{P}}_k$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img95.gif"
+ ALT="$ \tilde{\mathbb{P}}_k$"> of homogeneous
+ polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> is the span of monomials of total degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img96.gif"
+ ALT="$ d$"> variables on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">.
+</FONT>
+<P>
+<P>
+<DIV><B>D<SMALL>EFINITION</SMALL> 3</B>
+We define the auxiliary space
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="def:_space_Sk"></A><!-- MATH
+ \begin{equation}
+\mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d : \underline p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="317" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
+ SRC="img97.gif"
+ ALT="$\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d :...
+...ine p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(4)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+with <!-- MATH
+ $\hat{x} \in \hat{K}$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img98.gif"
+ ALT="$ \hat{x} \in \hat{K}$">.
+<BR>
+The dimension of this space is <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in the case <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> and <IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img99.gif"
+ ALT="$ k(k+2)$"> for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">.
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Nédélec's first family of <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite elements is based on the polynomial spaces
+ </FONT><P>
+<DIV><B>D<SMALL>EFINITION</SMALL> 4</B>
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="def:_space_Rk"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="196" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img100.gif"
+ ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(5)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+These spaces have dimension
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+ \mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \textrm{for} \quad d=2\,, \\
+ \mathrm{dim} (\mathcal{R}^k) &= \frac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,.
+ \end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="446" HEIGHT="72" BORDER="0"
+ SRC="img101.gif"
+ ALT="\begin{displaymath}\begin{split}\mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \t...
+...ac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">In the two-dimensional case, an equivalent characterization of the space <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> is
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_spaces_2d"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1
+ \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="293" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img103.gif"
+ ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus ...
+..._{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(6)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ This can be seen by noting that for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$">
+ </FONT><!-- MATH
+ \begin{displaymath}
+\tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \subseteq \mathcal{S}^k
+ %%\left\{\v p\quad \big|\quad \v p = \tilde{p} \left(\begin{array}{cc} x_2 \\-x_1 \end{array}\right)\,,\, \tilde{p}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="164" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img104.gif"
+ ALT="$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ ...
+...ilde{p} \left(\begin{array}{cc} x_2 \\ -x_1 \end{array}\right)\,,\, \tilde{p}
+$">
+</DIV><P></P><FONT SIZE="-1">
+ obviously holds.
+ Moreover, the dimension of the space <!-- MATH
+ $\tilde{\mathbb{P}}_{k-1}$
+ -->
+<IMG
+ WIDTH="43" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img105.gif"
+ ALT="$ \tilde{\mathbb{P}}_{k-1}$"> of homogeneous polynomials of degree <IMG
+ WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img106.gif"
+ ALT="$ k-1$"> in two variables is <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> and this is
+ also the dimension <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">. This proves the stated equivalent representation of the space <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">.
+<BR>
+<BR>
+We illustrate these definitions with some examples. We start with the case <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img13.gif"
+ ALT="$ d=2$"> and consider the spaces of polynomials of degree
+ <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$"> and <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img109.gif"
+ ALT="$ k=2$">:
+</FONT>
+<P>
+<P>
+<DIV><B>E<SMALL>XAMPLE</SMALL> 1</B>
+
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_R1"></A><!-- MATH
+ \begin{equation}
+\mathcal{R}^1 = \left\langle
+ \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,_,
+ \left(\begin{array}{cc} 0 \\1 \end{array}\right)\,_,
+ \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \right\rangle
+
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="278" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img110.gif"
+ ALT="$\displaystyle \mathcal{R}^1 = \left\langle \left(\begin{array}{cc} 1 \\ 0 \end...
+...eft(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right) \right\rangle$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(7)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+<!-- MATH
+ \begin{displaymath}
+\mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
+ \left\langle
+ \left(\begin{array}{cc} \hat{x}_1\,\hat{x}_2 \\-{\hat{x}_1}^2 \end{array}\right)\,_,
+ \left(\begin{array}{cc} {\hat{x}_2}^2 \\-\hat{x}_1\,\hat{x}_2 \end{array}\right)
+ \right\rangle
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img111.gif"
+ ALT="$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
+\...
+...{cc} {\hat{x}_2}^2 \\ -\hat{x}_1\,\hat{x}_2 \end{array}\right)
+\right\rangle
+$">
+</DIV><P></P>
+
+<P>
+</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">To illustrate a case for <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.gif"
+ ALT="$ d=3$">, we consider the lowest polynomial degree <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">:
+ </FONT><P>
+<DIV><B>E<SMALL>XAMPLE</SMALL> 2</B>
+We have to specify a basis for <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$">:
+<BR>
+Let <!-- MATH
+ $\underline p$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img113.gif"
+ ALT="$ \underline p$"> be a polynomial in <!-- MATH
+ $(\mathbb{P}_{1}(\hat{K}))^3$
+ -->
+<IMG
+ WIDTH="75" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img114.gif"
+ ALT="$ (\mathbb{P}_{1}(\hat{K}))^3$"> with componentwise representation
+ <!-- MATH
+ \begin{displaymath}
+p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="234" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img115.gif"
+ ALT="$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
+$">
+</DIV><P></P>
+The condition for <!-- MATH
+ $\underline p$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img113.gif"
+ ALT="$ \underline p$"> being in <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$"> is
+ <!-- MATH
+ \begin{displaymath}
+\underline p \cdot \hat{\underline x} =
+ \sum_{i=1}^3 a_{ii}\hat{x}_i^2 + \sum_{\substack{i,j=1 \\j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="327" HEIGHT="86" ALIGN="MIDDLE" BORDER="0"
+ SRC="img116.gif"
+ ALT="$\displaystyle \underline p \cdot \hat{\underline x} =
+\sum_{i=1}^3 a_{ii}\hat{...
+...um_{\substack{i,j=1 \\ j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
+$">
+</DIV><P></P>
+This leads to the condition on the coefficients of a polynomial in <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$">:
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+ &a_{11}=a_{22}=a_{33} = 0 \\
+ &a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,.
+ \end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="439" HEIGHT="55" BORDER="0"
+ SRC="img117.gif"
+ ALT="\begin{displaymath}\begin{split}&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}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+With the basis of <!-- MATH
+ $\mathcal{S}^1$
+ -->
+<IMG
+ WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img112.gif"
+ ALT="$ \mathcal{S}^1$"> which is obtained by choosing <!-- MATH
+ $a_{ij} = 1$
+ -->
+<IMG
+ WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img118.gif"
+ ALT="$ a_{ij} = 1$">, <IMG
+ WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img119.gif"
+ ALT="$ i=1,2,3$">, <IMG
+ WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img120.gif"
+ ALT="$ j>i$"> and setting all the other
+ coefficients to zero, we get
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus
+ \left\langle
+ \left(\begin{array}{ccc} 0 \\\hat{x}_3 \\\hat{x}_2 \end{array}\right)\,_,\,
+ \left(\begin{array}{ccc} \hat{x}_3 \\0 \\\hat{x}_1 \end{array}\right)\,_,\,
+ \left(\begin{array}{ccc} \hat{x}_2 \\\hat{x}_1 \\0 \end{array}\right)
+ \right\rangle
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="402" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img121.gif"
+ ALT="$\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus \l...
+...egin{array}{ccc} \hat{x}_2 \\ \hat{x}_1 \\ 0 \end{array}\right) \right\rangle$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+<P>
+</DIV><P></P>
+
+<P>
+<FONT SIZE="-1">We remark at this point that the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> do not span the whole <!-- MATH
+ $(\mathbb{P}_{k}(\hat{K}) )^d$
+ -->
+<IMG
+ WIDTH="76" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img122.gif"
+ ALT="$ (\mathbb{P}_{k}(\hat{K}) )^d$">.
+ An <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM based on full polynomial spaces, the so called
+ <I>Nédélec elements of second type</I>, was introduced in 1986 by Nédélec in [<A
+ HREF="node4.html#Ned2">10</A>].
+</FONT>
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL> 3</B>
+The original, rather technical, representation of the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> is given in Definition 2 in [<A
+ HREF="node4.html#Ned1">8</A>].
+ Nédélec uses this representation in most of his proofs. We will not refer to it here.
+ </DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00021200000000000000"></A>
+ <A NAME="subsubsect:_dofs_on_tria"></A>
+<BR>
+2.1.2 Degrees of freedom on the reference element
+</H3><FONT SIZE="-1">
+ In this section we define the set <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> of dofs, which is a set of linear functionals on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">.
+ </FONT><P>
+<DIV><B>R<SMALL>EMARK</SMALL> 4</B>
+Recall that the dimension of the spaces of polynomials of degree <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> in <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img123.gif"
+ ALT="$ n$"> variables is <!-- MATH
+ $n+k+2 \choose n$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img124.gif"
+ ALT="$ n+k+2 \choose n$">.
+ </DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_dofs_2d"><B>D<SMALL>EFINITION</SMALL> 5</B></A>
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference triangle and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>.
+ The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> in the 2d case consists of the linear functionals
+ <DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall \hat{\varphi }
+ \in \mathbb{P}_{k-1}(\hat{e})\,,
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img126.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img128.gif"
+ ALT="$ 3k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi }
+ \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img129.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
+...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img130.gif"
+ ALT="$ k(k-1)$"> of inner dofs.
+
+<P>
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_dofs_3d"><B>D<SMALL>EFINITION</SMALL> 6</B></A>
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference tetrahedron, <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A>
+ and <!-- MATH
+ $\hat{\underline n}$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img131.gif"
+ ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face.
+ The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> in the 3d case consists of the linear functionals
+ <DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall
+ \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img126.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of the tetrahedron <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img132.gif"
+ ALT="$ 6k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>face dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \quad
+ \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="335" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img133.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
+... }\,d\hat{a} \quad \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every face <IMG
+ WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img134.gif"
+ ALT="$ \hat{f}$"> of the tetrahedron <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. We have a total of <IMG
+ WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img135.gif"
+ ALT="$ 4k(k-1)$"> of face dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad
+ \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img136.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
+...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <!-- MATH
+ $\frac{k(k-1)(k-2)}{2}$
+ -->
+<IMG
+ WIDTH="88" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img137.gif"
+ ALT="$ \frac{k(k-1)(k-2)}{2}$"> of inner dofs.
+
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">We note that in the case of lowest order elements, i. e. <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$">, only edge dofs occur. This is not so for higher order elements. For
+ <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img109.gif"
+ ALT="$ k=2$"> we additionally have inner dofs in the 2d case and face dofs in the 3d case. For <IMG
+ WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img138.gif"
+ ALT="$ k\leq3$"> we have all types of dofs in both
+ cases.
+<BR>
+We also note that the total number of dofs equals the dimension of the spaces <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">, as it should be.
+<BR>
+The representation of the <I>interface</I> dofs, that is edge dofs in 2d, edge and face dofs in 3d, is motivated by the continuity
+ condition on <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-functions stated in proposition <A HREF="node1.html#prop:_no_jumps">1</A>.
+</FONT>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL> 2</B>
+The set <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> of dofs befined above is unisolvent on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">.
+ <!-- MATH
+ $\hat{\underline u}\in \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="61" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img139.gif"
+ ALT="$ \hat{\underline u}\in \mathcal{R}^k$"> is uniquely defined by the moments <!-- MATH
+ $\hat{\alpha}(\hat{\underline u})$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img140.gif"
+ ALT="$ \hat{\alpha}(\hat{\underline u})$">.
+ </DIV><P></P>
+<FONT SIZE="-1">
+ </FONT><FONT SIZE="-1">P<SMALL>ROOF. </SMALL>See [<A
+ HREF="node4.html#Ned1">8</A>], proof of theorem 1 and preceeding lemmas.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="ex:_shape_functions"><B>E<SMALL>XAMPLE</SMALL> 3</B></A> (Reference shape functions of lowest order for Nédélec elements on triangular meshes)
+Let the reference element be the triangle
+ <!-- MATH
+ $\hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$
+ -->
+<IMG
+ WIDTH="374" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img141.gif"
+ ALT="$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$">.
+ Label the edges couterclockwise startung with <!-- MATH
+ $\hat{e}_0 = \overline{(0,0),(1,0)}$
+ -->
+<IMG
+ WIDTH="131" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img142.gif"
+ ALT="$ \hat{e}_0 = \overline{(0,0),(1,0)}$">.
+ The tangential vectors to the edges are (oriented counterclockwise)
+ <!-- MATH
+ \begin{displaymath}
+\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
+ \hat{\underline t}_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} -1 \\1 \end{array}\right)\,,\quad
+ \hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="398" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img143.gif"
+ ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array...
+...
+\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,.
+$">
+</DIV><P></P>
+The underlying function space for lowest order Nédélec elements on a triangular mesh is <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$"> from (<A HREF="#eq:_____R1"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>).
+<BR>
+In the case of <IMG
+ WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img108.gif"
+ ALT="$ k=1$"> only egde-dofs occur. On <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> we have dofs of the type
+ <!-- MATH
+ $\int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="202" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img145.gif"
+ ALT="$ \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$">.
+ More precisely, since <!-- MATH
+ $\varphi \equiv 1$
+ -->
+<IMG
+ WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img146.gif"
+ ALT="$ \varphi \equiv 1$"> is a basis for <!-- MATH
+ $\mathbb{P}_{0}(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img147.gif"
+ ALT="$ \mathbb{P}_{0}(\hat{e}_i)$"> we have the three dofs
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="250" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img148.gif"
+ ALT="$\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+In order to construct a FE-basis <!-- MATH
+ $\hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img149.gif"
+ ALT="$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$"> for <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$"> with respect to these dofs, we require
+ <!-- MATH
+ $\hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$
+ -->
+<IMG
+ WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img150.gif"
+ ALT="$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$">. This leads to a linear system for
+ the coefficients of the <!-- MATH
+ $\hat{\underline N}_i$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img151.gif"
+ ALT="$ \hat{\underline N}_i$"> in a general basis of <!-- MATH
+ $\mathcal{R}^1$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img144.gif"
+ ALT="$ \mathcal{R}^1$">. In the case of lowest order elements, it is easy to
+ verify that we have
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\\hat{x} \end{array}\right)\,,\quad
+ \hat{\underline N}_1 = \left(\begin{array}{cc} -\hat{y} \\\hat{x} \end{array}\right)\,,\quad
+ \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\\hat{x}-1 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="446" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img152.gif"
+ ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ \hat...
+...ine N}_2 = \left(\begin{array}{cc} -\hat{y} \\ \hat{x}-1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(8)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00021300000000000000"></A> <A NAME="sec:_Piola_for_triangles"></A>
+<BR>
+2.1.3 Piola transformation
+</H3><FONT SIZE="-1">
+ An affine triangle or tetrahedron <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> is described by the affine element map
+ </FONT><!-- MATH
+ \begin{displaymath}
+K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img153.gif"
+ ALT="$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
+$">
+</DIV><P></P>
+<P>
+<FONT SIZE="-1">In standard <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$">-conforming FEM, the shape functions <IMG
+ WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img154.gif"
+ ALT="$ N_i$"> on a general cell <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> are obtained from the reference shape functions
+ <IMG
+ WIDTH="26" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img155.gif"
+ ALT="$ \hat{N}_i$"> on the reference element <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> by the pull-back
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="187" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
+ SRC="img156.gif"
+ ALT="$\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ In the case of <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming Nédélec FEM we cannot transforme our shape function in this way. The pull-back of a
+ <!-- MATH
+ $H(\mathop{\rm curl};\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img157.gif"
+ ALT="$ H(\mathop{\rm curl};\hat{K})$">-function needs not to be in <!-- MATH
+ $H(\mathop{\rm curl}; K)$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img158.gif"
+ ALT="$ H(\mathop{\rm curl}; K)$">. In addition, the pull-back is not an <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">-isomorphism and it
+ does not lead to an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method if prescribing the dofs by definitions <A HREF="node2.html#def:_dofs_2d">5</A> or <A HREF="#def:_dofs_____3d"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>.
+<BR>
+In Nédélec's FEM (or, more general, in <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM), the shape functions are transformed by the following covariant
+ transformation for vector-fields:
+</FONT>
+<P>
+<DL COMPACT>
+<DT></DT>
+<DD>The element shape functions <!-- MATH
+ $\underline N_i(x)$
+ -->
+<IMG
+ WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img159.gif"
+ ALT="$ \underline N_i(x)$"> on the element <!-- MATH
+ $K = F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$"> are obtained
+ from the reference shape functions by
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Piola"></A><!-- MATH
+ \begin{equation}
+\underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="329" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
+ SRC="img160.gif"
+ ALT="$\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(9)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+where <!-- MATH
+ $\hat{D}F_K$
+ -->
+<IMG
+ WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img161.gif"
+ ALT="$ \hat{D}F_K$"> is the jacobian <!-- MATH
+ $\frac{d}{d\hat{x}}F_K(\hat{x})$
+ -->
+<IMG
+ WIDTH="73" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img162.gif"
+ ALT="$ \frac{d}{d\hat{x}}F_K(\hat{x})$"> of the element map.
+
+</DD>
+</DL>
+<P>
+<FONT SIZE="-1">In literature, an equivalent to this transformation for <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$">-conforming FEM (which in that case is a contravariant map)
+ is referred to as <I>Piola transformation</I>, cf. [<A
+ HREF="node4.html#Brezzi-Fortin">3</A>] pp. 97.
+<BR>
+Here, we will refer to the transformation (<A HREF="node2.html#eq:_Piola">9</A>) of the vector field also as <I>Piola transformation</I>.
+<BR>
+We note that the gradients of scalar nodal <!-- MATH
+ $H^1(\Omega )$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img37.gif"
+ ALT="$ H^1(\Omega )$">-conforming finite elements transform according to the Piola
+ transformation (<A HREF="node2.html#eq:_Piola">9</A>).
+</FONT>
+<P>
+<FONT SIZE="-1">In the case of tetrahedral elements and affine element map <!-- MATH
+ $F_K(\hat{x}) = B_K \hat{x} + b_k$
+ -->
+<IMG
+ WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img164.gif"
+ ALT="$ F_K(\hat{x}) = B_K \hat{x} + b_k$">, the jacobian <!-- MATH
+ $\hat{D}F_K$
+ -->
+<IMG
+ WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img161.gif"
+ ALT="$ \hat{D}F_K$">
+ is just the constant matrix <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img165.gif"
+ ALT="$ B_K$"> and we have
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Piola_tri"></A><!-- MATH
+ \begin{equation}
+\underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="279" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img166.gif"
+ ALT="$\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(10)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+
+<H3><A NAME="SECTION00021400000000000000"></A>
+ <A NAME="subsubsect:_2d-curl_on_tria"></A>
+<BR>
+2.1.4 Transformation of the curl in 2d
+</H3><FONT SIZE="-1">
+ For <!-- MATH
+ $\Omega \subset\mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img167.gif"
+ ALT="$ \Omega \subset\mathbb{R}^2$">, we noted in remark <A HREF="node1.html#re:_rotation">1</A> that vector fields in <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$"> can be represented as
+ rotated <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$"> vector fields.
+ Moreover, it is easy to verify that
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_rot._jacobian"></A><!-- MATH
+ \begin{equation}
+B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img168.gif"
+ ALT="$\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(11)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ where <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img169.gif"
+ ALT="$ R$"> is the rotation matrix from remark <A HREF="node1.html#re:_rotation">1</A>.
+ Therefore, the properties of the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) in the 2d case can be derived directly
+ from the properties
+ of the <!-- MATH
+ $H(\mathop{\rm div}; \Omega )$
+ -->
+<IMG
+ WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img163.gif"
+ ALT="$ H(\mathop{\rm div}; \Omega )$">-Piola transformation stated in [<A
+ HREF="node4.html#Brezzi-Fortin">3</A>] pp. 97.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="th:_piola_2d"><B>T<SMALL>HEOREM</SMALL> 4</B></A> (Some properties of 2d Piola transformation for affine element map)
+Let <!-- MATH
+ $\underline v(x) = \mathcal{P}_K(\hat{\underline v})$
+ -->
+<IMG
+ WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img170.gif"
+ ALT="$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$">, <!-- MATH
+ $\varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$
+ -->
+<IMG
+ WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img171.gif"
+ ALT="$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$">, <!-- MATH
+ $\hat{x} =
+F_K^{-1}(x)$
+ -->
+<IMG
+ WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img172.gif"
+ ALT="$ \hat{x} =
+F_K^{-1}(x)$">, with affine element map <IMG
+ WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img173.gif"
+ ALT="$ F_K$">.
+ <DL COMPACT>
+<DT>(i)</DT>
+<DD>The gradient <!-- MATH
+ $D\underline v$
+ -->
+<IMG
+ WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img174.gif"
+ ALT="$ D\underline v$"> transforms according to
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_gradient_trafo_2d"></A><!-- MATH
+ \begin{equation}
+D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img175.gif"
+ ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(12)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+
+</DD>
+<DT>(ii)</DT>
+<DD>The curl transforms according to
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:curl_trafo_2d"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="176" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img176.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(13)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+As a consequence we see that <!-- MATH
+ $H(\mathop{\rm curl};K)$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img158.gif"
+ ALT="$ H(\mathop{\rm curl}; K)$"> is isomorphic to <!-- MATH
+ $H(\mathop{\rm curl};\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img157.gif"
+ ALT="$ H(\mathop{\rm curl};\hat{K})$"> under the Piola transformation
+ (<A HREF="node2.html#eq:_Piola_tri">10</A>).
+
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ </FONT><DL COMPACT>
+<DT>(i)</DT>
+<DD>Chain rule
+
+</DD>
+<DT>(ii)</DT>
+<DD>We use that the 2d <!-- MATH
+ $\mathop{\rm curl}$
+ -->
+<IMG
+ WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img19.gif"
+ ALT="$ \mathop{\rm curl}$"> operator is just the trace of the rotated jacobian <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img177.gif"
+ ALT="$ R\,Dv$">. By remark <A HREF="node2.html#eq:_rot._jacobian">11</A>,
+ we can replace <IMG
+ WIDTH="42" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img178.gif"
+ ALT="$ B_K^{-T}$"> and we get that <IMG
+ WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img177.gif"
+ ALT="$ R\,Dv$"> is affine-equivalent to <!-- MATH
+ $\det B_K^{-1}\,R\,\hat{D}\hat{v}$
+ -->
+<IMG
+ WIDTH="109" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img179.gif"
+ ALT="$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$">,
+ which proves (ii).
+
+</DD>
+</DL><FONT SIZE="-1">
+
+</FONT>
+<P>
+<P>
+<DIV><B>C<SMALL>OROLLARY</SMALL> 1</B>
+From (ii) in theorem <A HREF="node2.html#th:_piola_2d">4</A> we deduce
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="240" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img180.gif"
+ ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\h...
+...}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+and we have, together with (ii) from theorem <A HREF="node2.html#th:_piola_2d">4</A>
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underline u \, dx = | B_K |^{-1}\,\int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\,
+ \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="362" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img181.gif"
+ ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underl...
+...hat{\underline v}\, \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<P>
+
+<H3><A NAME="SECTION00021500000000000000"></A>
+<A NAME="subsubsect:_3d-curl_on_tria"></A>
+<BR>
+2.1.5 Transformation of the curl in 3d
+</H3><FONT SIZE="-1">
+ In three dimensions, we cannot identify the curl-operator with the rotated
+ gradient or with the divergence of a rotated vector field. We cannot, as in 2d, derive a transformation formula for the curl from the
+ transformatin formula of the divergence.
+<BR>
+By the chain rule, we obtain the transformation of the gradient of a vector field <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">, defined by the Piola transformation (<A HREF="#eq:__Piola_tri"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>) of a reference field <!-- MATH
+ $\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img182.gif"
+ ALT="$ \hat{\underline v}$">:
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_trafo_grad_3d"></A><!-- MATH
+ \begin{equation}
+D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img175.gif"
+ ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(14)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We introduce the skew symmetric matrix <!-- MATH
+ $\mathop{\rm Curl}v$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img183.gif"
+ ALT="$ \mathop{\rm Curl}v$"> as
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Curl"></A><!-- MATH
+ \begin{equation}
+\left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="183" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img184.gif"
+ ALT="$\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(15)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We see that <!-- MATH
+ $\mathop{\rm Curl}v = D\underline v^T - D\underline v$
+ -->
+<IMG
+ WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img185.gif"
+ ALT="$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$"> and therefore by (<A HREF="node2.html#eq:_trafo_grad_3d">14</A>)
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Curl_trafo"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="192" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img186.gif"
+ ALT="$\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(16)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL> 3</B> (Transformation of the curl in 3d)
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference tetrahedron and <!-- MATH
+ $K=F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$"> an affine image of it.
+ The curl of a vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the Piola transformation of a reference field <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$">
+ transforms according to
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl_trafo_3d"></A><!-- MATH
+ \begin{equation}
+\left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="307" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img189.gif"
+ ALT="$\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(17)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We obtain the matrix <!-- MATH
+ $\mathrm{M_i}$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img190.gif"
+ ALT="$ \mathrm{M_i}$"> by replacing i-th column of the (constant) jacobian <!-- MATH
+ $D(F_K^{-1}) = B_K^{-1}$
+ -->
+<IMG
+ WIDTH="122" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img191.gif"
+ ALT="$ D(F_K^{-1}) = B_K^{-1}$">
+ by the vector <!-- MATH
+ $(\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$
+ -->
+<IMG
+ WIDTH="133" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img192.gif"
+ ALT="$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$">:
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll}
+ (\widehat{\mathop{\rm curl}}\,\hat{v} \circ F_K^{-1})_k (x) & \textrm{if} \quad l=i \\
+ (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i
+ \end{array} \right.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="351" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
+ SRC="img193.gif"
+ ALT="$\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll} (\...
+...} \quad l=i \\ (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i \end{array} \right.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+ (Note: an alternative, equivalent, transformation formula for the curl in 3d is given in proposition <A HREF="node2.html#prop:_Demko_curl">4</A>).
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ It holds
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl2Curl"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\mathop{\rm Curl}v})_{23} \\({\mathop{\rm Curl}v})_{31} \\({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="202" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img194.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\math...
+...\mathop{\rm Curl}v})_{31} \\ ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(18)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ We demonstrate the statement of the proposition for the first component of the curl, which is <!-- MATH
+ $(\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$
+ -->
+<IMG
+ WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img195.gif"
+ ALT="$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$">.
+ Using the transformation (<A HREF="node2.html#eq:_Curl_trafo">16</A>), implicit summation over equal indices and the
+ abbreviation <!-- MATH
+ $b_{ij} := (B_K^{-1})_{ij}$
+ -->
+<IMG
+ WIDTH="111" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img196.gif"
+ ALT="$ b_{ij} := (B_K^{-1})_{ij}$">, we have
+ </FONT><!-- MATH
+ \begin{displaymath}
+({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="223" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img197.gif"
+ ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
+$">
+</DIV><P></P><FONT SIZE="-1">
+ Writing this out and recalling that <!-- MATH
+ $\mathop{\rm Curl}v$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img183.gif"
+ ALT="$ \mathop{\rm Curl}v$"> is skew symmetric, yields
+ </FONT><!-- MATH
+ \begin{displaymath}
+({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\widehat{\mathop{\rm Curl}} \,\hat{v})_{12}
+ -(b_{12}b_{33} - b_{32}b_{13})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{31}
+ +(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="739" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img198.gif"
+ ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\wide...
+...
++(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ and with (<A HREF="node2.html#eq:_curl2Curl">18</A>) this is equal to the determinant of
+ </FONT><!-- MATH
+ \begin{displaymath}
+\mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm curl}}\, v)_1 & b_{12} & b_{13} \\
+ (\widehat{\mathop{\rm curl}}\, v)_2 & b_{22} & b_{23} \\
+ (\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33}
+ \end{array}\right) \,.
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="253" HEIGHT="93" ALIGN="MIDDLE" BORDER="0"
+ SRC="img199.gif"
+ ALT="$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm cu...
+...
+(\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33}
+\end{array}\right) \,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+ The proof for the other components follows analogously.
+
+</FONT>
+<P>
+<FONT SIZE="-1">In the next proposition, we state an alternative, equivalent, formula for the transformation of the curl
+ (e. g. used by Demkovicz in [<A
+ HREF="node4.html#Demko3d">12</A>])
+ </FONT><P>
+<DIV><A NAME="prop:_Demko_curl"><B>P<SMALL>ROPOSITION</SMALL> 4</B></A>
+For a vector field <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$"> on the tetrahedron <!-- MATH
+ $K=F_K(\hat{K})$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img83.gif"
+ ALT="$ K = F_K(\hat{K})$">, defined by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) of a reference
+ field <!-- MATH
+ $\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img182.gif"
+ ALT="$ \hat{\underline v}$"> on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">, we have
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_3d_curl_trafo_Demko"></A><!-- MATH
+ \begin{equation}
+\mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="271" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img200.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(19)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ The transformation formula (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) can be proven componentwise, and
+ we will only carry out the proof for the first vector component <!-- MATH
+ $(\mathop{\rm curl}\underline v)_1$
+ -->
+<IMG
+ WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img201.gif"
+ ALT="$ (\mathop{\rm curl}\underline v)_1$">. The proofs for the other components
+ follow analogously.
+<BR>
+The identity (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) reads for the first vector component
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_1.comp_of_curl_trafo"></A><!-- MATH
+ \begin{equation}
+(\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="334" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img202.gif"
+ ALT="$\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(20)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ Referring to (<A HREF="node2.html#eq:_curl_trafo_3d">17</A>), we show that the right hand side of (<A HREF="node2.html#eq:_1.comp_of_curl_trafo">20</A>) equals
+ <!-- MATH
+ $\det \mathrm{M_1}$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img203.gif"
+ ALT="$ \det \mathrm{M_1}$">.
+ For this, we expand <!-- MATH
+ $\det \mathrm{M_1}$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img203.gif"
+ ALT="$ \det \mathrm{M_1}$"> to
+ </FONT><!-- MATH
+ \begin{displaymath}
+\det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det\mathcal{B}^{inv}_{11}
+ -(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det\mathcal{B}^{inv}_{21}
+ +(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="505" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img204.gif"
+ ALT="$\displaystyle \det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underlin...
+...ehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ where <!-- MATH
+ $\mathcal{B}^{inv}_{ij}$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img205.gif"
+ ALT="$ \mathcal{B}^{inv}_{ij}$"> is the <!-- MATH
+ $2 \times 2$
+ -->
+<IMG
+ WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img206.gif"
+ ALT="$ 2 \times 2$">-matrix arising from <IMG
+ WIDTH="39" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img207.gif"
+ ALT="$ B_K^{-1}$"> when cancelling its i-th row and its j-th column.
+<BR>
+We recall the formula for the inverse of a matrix <!-- MATH
+ $A \in \mathbb{R}^{3\times 3}$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img208.gif"
+ ALT="$ A \in \mathbb{R}^{3\times 3}$">
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_Ainv"></A><!-- MATH
+ \begin{equation}
+(A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="251" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
+ SRC="img209.gif"
+ ALT="$\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(21)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ where <!-- MATH
+ $\mathcal{A}_{ij}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img210.gif"
+ ALT="$ \mathcal{A}_{ij}$"> is the <!-- MATH
+ $2 \times 2$
+ -->
+<IMG
+ WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img206.gif"
+ ALT="$ 2 \times 2$">-matrix arising from <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img211.gif"
+ ALT="$ A$"> when cancelling its i-th row and its j-th column.
+<BR>
+Replacing <IMG
+ WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img165.gif"
+ ALT="$ B_K$"> in the right hand side of (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) by the expression (<A HREF="node2.html#eq:_Ainv">21</A>)
+ for <!-- MATH
+ $A = B_K^{-1}$
+ -->
+<IMG
+ WIDTH="75" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img212.gif"
+ ALT="$ A = B_K^{-1}$">, we get
+ </FONT><!-- MATH
+ \begin{displaymath}
+\frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \mathcal{B}^{inv}_{j1} (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_j
+ = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det \mathcal{B}^{inv}_{11}
+ -(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det \mathcal{B}^{inv}_{21}
+ +(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,.
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="834" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
+ SRC="img213.gif"
+ ALT="$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \math...
+...l}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00022000000000000000"></A> <A NAME="sec:_parallelo"></A>
+<BR>
+2.2 Nédélec Elements on affine quadrilateral or hexahedral grids
+</H2><FONT SIZE="-1">
+ We want to present the ingredients for Nédélec's finite elements of first type on
+ grids consisiting of parallelograms (in 2d) or the respective objects in 3d, so called parallelotops (cf. section
+ <I>FE built on cubes</I> in [<A
+ HREF="node4.html#Ned1">8</A>]). Such grids consist
+ of elements <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> that are affine images of the square or cubic reference element <!-- MATH
+ $\hat{C} = [0,1]^d$
+ -->
+<IMG
+ WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img215.gif"
+ ALT="$ \hat{C} = [0,1]^d$">:
+ </FONT><!-- MATH
+ \begin{displaymath}
+C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img216.gif"
+ ALT="$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
+$">
+</DIV><P></P>
+<P>
+
+<H3><A NAME="SECTION00022100000000000000">
+2.2.1 Polynomial spaces on the reference element</A>
+</H3>
+<P>
+<FONT SIZE="-1">In order to introduce the function spaces needed for the construction of Nédélec's finite elements,
+ let us define some spaces of vector-valued polynomials
+ </FONT><P>
+<DIV><B>D<SMALL>EFINITION</SMALL> 7</B>
+<!-- MATH
+ $\mathcal{Q}_{l,m}$
+ -->
+<IMG
+ WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img217.gif"
+ ALT="$ \mathcal{Q}_{l,m}$"> are the spaces of polynomials on the reference square <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> with maximal degree <IMG
+ WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img219.gif"
+ ALT="$ l$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img220.gif"
+ ALT="$ \hat{x}_1$"> and
+ <IMG
+ WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img221.gif"
+ ALT="$ m$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img222.gif"
+ ALT="$ \hat{x}_2$">.
+<BR> <!-- MATH
+ $\mathcal{Q}_{l,m,n}$
+ -->
+<IMG
+ WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img223.gif"
+ ALT="$ \mathcal{Q}_{l,m,n}$"> are the spaces of polynomials on the reference cube <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> with maximal degree <IMG
+ WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img219.gif"
+ ALT="$ l$">
+ in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img220.gif"
+ ALT="$ \hat{x}_1$">, <IMG
+ WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img221.gif"
+ ALT="$ m$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img222.gif"
+ ALT="$ \hat{x}_2$"> and <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img123.gif"
+ ALT="$ n$"> in <IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img224.gif"
+ ALT="$ \hat{x}_3$">.
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">The spaces <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img84.gif"
+ ALT="$ \hat{R}$"> for the reference shape functions now are in 2d
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_2d_spaces_for_quads"></A><!-- MATH
+ \begin{equation}
+\mathcal{P}^k = \left\{ \hat{\underline u} =
+ \left(\begin{array}{cc} \hat{u}_1 \\\hat{u}_2 \end{array}\right): \quad
+ \hat{u}_1 \in \mathcal{Q}_{k-1,k}\,,
+ \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,
+
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="402" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img225.gif"
+ ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
+...t{u}_1 \in \mathcal{Q}_{k-1,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(22)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+</FONT>
+<P>
+<FONT SIZE="-1">and in 3d
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_3d_spaces_for_quads"></A><!-- MATH
+ \begin{equation}
+\mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{ccc}
+ \hat{u}_1 \\\hat{u}_2 \\\hat{u}_3 \end{array}\right):\quad
+ \hat{u}_1 \in \mathcal{Q}_{k-1,k,k}\,,
+ \hat{u}_2 \in \mathcal{Q}_{k,k-1,k}\,,
+ \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.
+
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="546" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
+ SRC="img226.gif"
+ ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
+...}_2 \in \mathcal{Q}_{k,k-1,k}\,, \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(23)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+</FONT>
+<P>
+<FONT SIZE="-1">We renounce an example, since it is quite evident, what these spaces look like for a specific <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$">.
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00022200000000000000"></A>
+<A NAME="subsubsect:_dofs_on_quads"></A>
+<BR>
+2.2.2 Degrees of freedom on the reference element
+</H3><FONT SIZE="-1">
+ We start with the degrees of freedoms on the reference square <!-- MATH
+ $\hat{C}\subset \mathbb{R}^2$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img227.gif"
+ ALT="$ \hat{C}\subset \mathbb{R}^2$">:
+ </FONT><P>
+<DIV><A NAME="def:_quad-dofs_2d"><B>D<SMALL>EFINITION</SMALL> 8</B></A>
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> denote the reference square and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>.
+ The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> in the 2d case consists of the linear functionals
+ <DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
+ \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img229.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <IMG
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img230.gif"
+ ALT="$ 4k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x}\,, \quad
+ \forall\, \hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
+ \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="566" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img231.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
+... }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img232.gif"
+ ALT="$ 2k(k-1)$"> of inner dofs.
+
+<P>
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><A NAME="def:_quad-dofs_3d"><B>D<SMALL>EFINITION</SMALL> 9</B></A>
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> denote the reference cube, <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A>
+ and <!-- MATH
+ $\hat{\underline n}$
+ -->
+<IMG
+ WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img131.gif"
+ ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face.
+ The set of degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> on <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> in the 3d case consists of the linear functionals
+ <DL>
+<DT><STRONG>edge dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
+ \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img229.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
+... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <IMG
+ WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img233.gif"
+ ALT="$ 12k$"> of edge dofs.
+
+<P>
+</DD>
+<DT><STRONG>face dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \,,\quad
+ \forall \,\hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
+ \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="651" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img234.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
+..._{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for every face <IMG
+ WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img134.gif"
+ ALT="$ \hat{f}$"> of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. We have a total of <!-- MATH
+ $6\cdot 2k(k-1)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img235.gif"
+ ALT="$ 6\cdot 2k(k-1)$"> of face dofs.
+
+<P>
+</DD>
+<DT><STRONG>inner dofs</STRONG></DT>
+<DD><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \,,\quad
+ \forall\, \hat{\underline \varphi } = \left(\begin{array}{ccc} \hat{\varphi }_1 \\\hat{\varphi }_2 \\
+ \hat{\varphi }_3\end{array}\right) \,,\quad\hat{\varphi }_1\in\mathcal{Q}_{k-1,k-2,k-2}\,,\quad\hat{\varphi _2}\in\mathcal{Q}_{k-2,k-1,k-2}\,,
+ \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="793" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
+ SRC="img236.gif"
+ ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
+...athcal{Q}_{k-2,k-1,k-2}\,, \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+We have a total of <IMG
+ WIDTH="86" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img237.gif"
+ ALT="$ 3k(k-1)^2$"> of inner dofs.
+
+</DD>
+</DL></DIV><P></P>
+
+<P>
+<P>
+<DIV><B>E<SMALL>XAMPLE</SMALL> 4</B>
+Proceeding the same way as in example <A HREF="node2.html#ex:_shape_functions">3</A> for a triangular reference element, we obtain
+the reference shape functions of lowest
+order on the square <IMG
+ WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img238.gif"
+ ALT="$ [0,1]^2$">. For the unit tangents as in convention <A HREF="node1.html#def:_tangent">1</A>
+<!-- MATH
+ \begin{displaymath}
+\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
+ \hat{\underline t}_1 = \left(\begin{array}{cc} 0 \\1 \end{array}\right)\,,\quad
+ \hat{\underline t}_2 = \left(\begin{array}{cc} -1 \\0 \end{array}\right)\,, \quad
+ \hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="487" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="img239.gif"
+ ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array...
+...
+\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,,
+$">
+</DIV><P></P>
+they read
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\0 \end{array}\right)\,,\quad
+ \hat{\underline N}_1 = \left(\begin{array}{cc} 0 \\\hat{x} \end{array}\right)\,,\quad
+ \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\0 \end{array}\right)\,,\quad
+ \hat{\underline N}_3 = \left(\begin{array}{cc} 0 \\\hat{x}-1 \end{array}\right)\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="573" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img240.gif"
+ ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ 0 \e...
+...\underline N}_3 = \left(\begin{array}{cc} 0 \\ \hat{x}-1 \end{array}\right)\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(24)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+
+<H3><A NAME="SECTION00022300000000000000">
+2.2.3 Transformation of the vector field</A>
+</H3><FONT SIZE="-1">
+Since the elements of the considered grids are still affine images of the reference element, we can use the Piola transformation
+(<A HREF="node2.html#eq:_Piola_tri">10</A>) to transform vector fields and the results stated in sections
+<A HREF="node2.html#sec:_Piola_for_triangles">2.1.3</A> - <A HREF="node2.html#subsubsect:_3d-curl_on_tria">2.1.5</A> can be carried over one to one.
+</FONT>
+<H2><A NAME="SECTION00023000000000000000">
+2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
+</H2><FONT SIZE="-1">
+ We now want to consider grids that are composed of elements that are a bi- resp. trilinear images <!-- MATH
+ $F_C(\hat{C})$
+ -->
+<IMG
+ WIDTH="56" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img241.gif"
+ ALT="$ F_C(\hat{C})$"> of the reference
+ element <!-- MATH
+ $\hat{C} = [0,1]^d$
+ -->
+<IMG
+ WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img215.gif"
+ ALT="$ \hat{C} = [0,1]^d$">.
+ The main difference here is, that the jacobian <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> of the element map <IMG
+ WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img243.gif"
+ ALT="$ F_C$"> is not
+ constant, and we have to use Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) to
+ transform vector fields.
+<BR></FONT>
+<P>
+
+<H3><A NAME="SECTION00023100000000000000">
+2.3.1 Bilinear elements in 2d</A>
+</H3><FONT SIZE="-1">
+ The polynomial spaces <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine quadrilateral elements.
+<BR>
+A transformed vector field on a general element is now defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>)
+ </FONT><!-- MATH
+ \begin{displaymath}
+\underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="212" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img244.gif"
+ ALT="$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$">
+</DIV><P></P><FONT SIZE="-1"> of a vector field on the reference element. Note that
+ the jacobian <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> is not constant in this case. In contrast to the case of affine elements, the gradient <!-- MATH
+ $D\underline v$
+ -->
+<IMG
+ WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img174.gif"
+ ALT="$ D\underline v$"> does not transform according to
+ formula (<A HREF="node2.html#eq:_gradient_trafo_2d">12</A>). Non-vanishing second derivatives of <!-- MATH
+ $\hat{D}F_C(\hat{x})$
+ -->
+<IMG
+ WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img242.gif"
+ ALT="$ \hat{D}F_C(\hat{x})$"> appear in the transformation rule
+ for gradients of vector fields.
+ This requires a new approach to express <!-- MATH
+ $\mathop{\rm curl}\underline v$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img245.gif"
+ ALT="$ \mathop{\rm curl}\underline v$"> in terms of
+ <!-- MATH
+ $\widehat{\mathop{\rm curl}}\,\hat{\underline v}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img246.gif"
+ ALT="$ \widehat{\mathop{\rm curl}}\,\hat{\underline v}$">.
+ Nevertheless, it can be shown that the curl of a vector field transforms analogously to the case of affine elements.
+</FONT>
+<P>
+<P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL> 5</B>
+Let <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$"> be the reference element <IMG
+ WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img238.gif"
+ ALT="$ [0,1]^2$"> and <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> a bilinear image of <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. If the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$">
+ transforms according to the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>), then the transformation of the curl obeys
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="370" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img247.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+as in the affine case.
+<BR> </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ In this proof, the mapped element <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img214.gif"
+ ALT="$ C$"> will be fixed, so for simplicity we write <IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img248.gif"
+ ALT="$ F$"> for <IMG
+ WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img243.gif"
+ ALT="$ F_C$">.
+<BR>
+First note that <!-- MATH
+ $(\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$
+ -->
+<IMG
+ WIDTH="242" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img249.gif"
+ ALT="$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$">. We use the notation <!-- MATH
+ $D(F^{-1})_{ij}(x)=
+\frac{\partial \hat{x}_i}{\partial x_j}(x)$
+ -->
+<IMG
+ WIDTH="174" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img250.gif"
+ ALT="$ D(F^{-1})_{ij}(x)=
+\frac{\partial \hat{x}_i}{\partial x_j}(x)$"> and imlicit summation to rewrite the Piola transformation of the vector field componentwise
+ </FONT><!-- MATH
+ \begin{displaymath}
+v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img251.gif"
+ ALT="$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+ In the case of affine elements, i. e. for constant jacobian, we have
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+ \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
+ \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,,
+ \end{split}
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="398" HEIGHT="90" BORDER="0"
+ SRC="img252.gif"
+ ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part...
+... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">whereas for non-constant jacobian we have
+</FONT>
+<P>
+<P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\begin{split}
+ \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) +
+ \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
+ \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) +
+ \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,.
+ \end{split}
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="493" HEIGHT="95" BORDER="0"
+ SRC="img253.gif"
+ ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part...
+... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath}"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+<P>
+<FONT SIZE="-1">We see that in <I>both</I> cases we have
+ </FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1} - \frac{\partial v_1}{\partial x_2} =
+ \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x))
+ - \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="548" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img254.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1}...
+...ial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ that is, the second derivatives cancel out in the expression for the curl and the curl in the non-affine case transforms equally
+ to the curl in the affine case.
+
+</FONT>
+<P>
+
+<H3><A NAME="SECTION00023200000000000000">
+2.3.2 Trilinear elements in 3d</A>
+</H3><FONT SIZE="-1">
+ The polynomial spaces <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine hexahedral elements.
+<BR>
+The
+ vector field on a genereal element is defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>).
+<BR>
+The problem of the non-vanishing second derivatives of the jacobian <!-- MATH
+ $D(F_C^{-1})(x)$
+ -->
+<IMG
+ WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img255.gif"
+ ALT="$ D(F_C^{-1})(x)$"> arises again, and we cannot generalize the results
+ from the affine case straight away.
+<BR>
+But analogously to the 2d case, one can check that in in the transformation rule for expressions
+ <!-- MATH
+ $\frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img256.gif"
+ ALT="$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$">, <!-- MATH
+ $i,j = 1,2,3$
+ -->
+<IMG
+ WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ SRC="img257.gif"
+ ALT="$ i,j = 1,2,3$">,
+ which define the curl-operator, the terms containing second derivatives vanish. We have therefore again the transformation rule
+ (<A HREF="node2.html#eq:_Curl_trafo">16</A>) for the skew matrix <!-- MATH
+ $\mathop{\rm Curl}v = Dv^T - Dv$
+ -->
+<IMG
+ WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img258.gif"
+ ALT="$ \mathop{\rm Curl}v = Dv^T - Dv$">:
+ </FONT><!-- MATH
+ \begin{displaymath}
+\mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,\hat{D}F_C^{-1}) \circ F_C^{-1})(x)
+ = (DF_C^{-1})^T(x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="673" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img259.gif"
+ ALT="$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm C...
+...x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
+$">
+</DIV><P></P>
+<P>
+<FONT SIZE="-1">It follows that the following proposition can be proved analogously to the case of affine elements (replace there <IMG
+ WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img260.gif"
+ ALT="$ B_C$"> by
+ <!-- MATH
+ $\hat{D}F_C(\hat(x))$
+ -->
+<IMG
+ WIDTH="81" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
+ SRC="img261.gif"
+ ALT="$ \hat{D}F_C(\hat(x))$">).
+ </FONT><P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL> 6</B>
+Let the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on a trilinear image <!-- MATH
+ $C = F_C(\hat{C})$
+ -->
+<IMG
+ WIDTH="93" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img262.gif"
+ ALT="$ C = F_C(\hat{C})$"> be defined by the Piola transformation of a
+ reference field <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
+ WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img218.gif"
+ ALT="$ \hat{C}$">. The transformation formula for the curl then reads
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\mathop{\rm curl}\underline v =
+ \left(\frac{1}{\det \hat{D}F_C}\,\hat{D}F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.
+
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="305" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
+ SRC="img263.gif"
+ ALT="$\displaystyle \mathop{\rm curl}\underline v = \left(\frac{1}{\det \hat{D}F_C}\,...
+...F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00024000000000000000">
+2.4 Construction of global shape functions</A>
+</H2><FONT SIZE="-1">
+ In the previous sections we have introduced function spaces and degrees of freedom, which, together with the Piola transformation,
+ will allow us to define an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method.
+ Indeed, in [<A
+ HREF="node4.html#Ned1">8</A>], Nédélec shows the invariance of the spaces
+ <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> and <!-- MATH
+ $\mathcal{Q}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img264.gif"
+ ALT="$ \mathcal{Q}^k$"> under Piola transformation of the vector field, as well as the unisolvence of the set of
+ degrees of freedom <!-- MATH
+ $\mathcal{A}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img89.gif"
+ ALT="$ \mathcal{A}$"> from sections <A HREF="node2.html#subsubsect:_dofs_on_tria">2.1.2</A> and <A HREF="node2.html#subsubsect:_dofs_on_quads">2.2.2</A>
+ (for details, see [<A
+ HREF="node4.html#Ned1">8</A>], Section 1.2, Theorem 1 and Section 2, Theorem 5).
+ This leads to the fact that <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions can be defined by mapping elementwise the reference
+ shape functions with the Piola transformation <!-- MATH
+ $\mathcal{P}_K$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img265.gif"
+ ALT="$ \mathcal{P}_K$">.
+ However, we must pay some care to the orientation of an interface on which the moments defining the degrees of freedom are based.
+ For the 2d case, we will illustrate in this section how we must take into account the <I>orientation of an edge</I>
+ in the definition of the respective element edge shape function, in order to get an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element space
+ of global shape functions.
+
+<BR></FONT>
+<P>
+<FONT SIZE="-1">Let <!-- MATH
+ $K=F(\hat{K})$
+ -->
+<IMG
+ WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img266.gif"
+ ALT="$ K=F(\hat{K})$"> be an affine or bilinear image of a reference element, <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> one of its edges and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> the
+ corresponding edge on the reference element.
+<BR>
+Let further <!-- MATH
+ $[0,|e|] \ni s \mapsto \underline x(s) \in e$
+ -->
+<IMG
+ WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img267.gif"
+ ALT="$ [0,\vert e\vert] \ni s \mapsto \underline x(s) \in e$"> and <!-- MATH
+ $[0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$
+ -->
+<IMG
+ WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img268.gif"
+ ALT="$ [0,\vert\hat{e}\vert] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$"> be parametrizations
+ with respect to the arc length of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> respectively. We can assume that these parametrizations endow the edges with a
+ counterclockwise orientation. Then, the unit tangent vectors <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$"> and <!-- MATH
+ $\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img125.gif"
+ ALT="$ \hat{\underline t}$"> are given by
+ <!-- MATH
+ $\frac{d \underline x}{ds}$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
+ SRC="img269.gif"
+ ALT="$ \frac{d \underline x}{ds}$"> and <!-- MATH
+ $\frac{d \hat{\underline x}}{d\hat{s}}$
+ -->
+<IMG
+ WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
+ SRC="img270.gif"
+ ALT="$ \frac{d \hat{\underline x}}{d\hat{s}}$">.
+</FONT>
+<P>
+<P>
+<DIV><A NAME="lemma:__v_dot_t_-Trafo"><B>L<SMALL>EMMA</SMALL> 1</B></A>
+Let <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> be a vector field on the reference element and <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> be the corresponding vector field on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the
+ Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>). It then holds
+ <P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation}
+\underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
+ SRC="img271.gif"
+ ALT="$\displaystyle \underline v\cdot \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert} (\hat{\underline v}\cdot \hat{\underline t})\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(25)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+where <IMG
+ WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img272.gif"
+ ALT="$ \vert\hat{e}\vert$"> and <IMG
+ WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img273.gif"
+ ALT="$ \vert e\vert$"> denote the length of the edges <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">.
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ With
+ </FONT><!-- MATH
+ \begin{displaymath}
+(\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="300" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
+ SRC="img274.gif"
+ ALT="$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ and <!-- MATH
+ $\hat{x}_j = \hat{x}_j(\underline x(s))$
+ -->
+<IMG
+ WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img275.gif"
+ ALT="$ \hat{x}_j = \hat{x}_j(\underline x(s))$"> and <!-- MATH
+ $\hat{x}_j = \hat{x}_j(\hat{s}(s))$
+ -->
+<IMG
+ WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img276.gif"
+ ALT="$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$"> on the edges, we have
+ </FONT><!-- MATH
+ \begin{displaymath}
+\underline v\cdot \underline t= \underline v \cdot \frac{d \underline x}{ds} = \left( \hat{\underline v}_j\frac{\partial \hat{x}_j}{\partial x_i}\right) (x)\frac{dx_i}{ds}
+ = \hat{\underline v}_j \frac{d \hat{x}_j}{ds} = \hat{\underline v}_j \frac{d \hat{x}_j}{d\hat{s}} \frac{d\hat{s}}{ds}
+ = (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds}
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="498" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
+ SRC="img277.gif"
+ ALT="$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \under...
+...t{s}}{ds}
+= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds}
+$">
+</DIV><P></P><FONT SIZE="-1">
+ and with <!-- MATH
+ $\frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$
+ -->
+<IMG
+ WIDTH="64" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img278.gif"
+ ALT="$ \frac{d\hat{s}}{ds}=\frac{\vert\hat{e}\vert}{\vert e\vert}$"> the lemma follows.
+
+</FONT>
+<P>
+<FONT SIZE="-1">As a consequence, we have
+</FONT>
+<P>
+<P>
+<DIV><A NAME="prop:_Dof_invariance"><B>P<SMALL>ROPOSITION</SMALL> 7</B></A> (Invariance of the edge dofs)
+Let the vector field <!-- MATH
+ $\underline v(x)$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img187.gif"
+ ALT="$ \underline v(x)$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> be defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) of a reference vector field
+ <!-- MATH
+ $\hat{\underline v}(\hat{x})$
+ -->
+<IMG
+ WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img188.gif"
+ ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$">. Then, the functionals (<I>edge dofs</I>)
+ <!-- MATH
+ $\alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$
+ -->
+<IMG
+ WIDTH="178" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="img279.gif"
+ ALT="$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$"> are invariant in the sense of
+ <!-- MATH
+ \begin{displaymath}
+\alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline t)\varphi \,ds =
+ \int_{\hat{e}} (\hat{\underline v} \cdot \hat{\underline t}) \hat{\varphi } \, d\hat{s}\, = \hat{\alpha}(\hat{\underline u})\,,
+ \qquad \forall\, \hat{\varphi } \in
+ \mathbb{P}_{k-1}(\hat{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="704" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
+ SRC="img280.gif"
+ ALT="$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline...
+...{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">Let now <!-- MATH
+ $K_- = F_-(\hat{K})$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img281.gif"
+ ALT="$ K_- = F_-(\hat{K})$"> and <!-- MATH
+ $K_+ = F_+(\hat{K})$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img282.gif"
+ ALT="$ K_+ = F_+(\hat{K})$"> be two neighbouring triangles with common edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">. Let <!-- MATH
+ $\underline N$
+ -->
+<IMG
+ WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img283.gif"
+ ALT="$ \underline N$"> be the global edge shape
+ function that 'lives' on <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">. By <!-- MATH
+ $\underline N_-$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img284.gif"
+ ALT="$ \underline N_-$"> and <!-- MATH
+ $\underline N_+$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img285.gif"
+ ALT="$ \underline N_+$"> we denote the restriction of <!-- MATH
+ $\underline N$
+ -->
+<IMG
+ WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img283.gif"
+ ALT="$ \underline N$"> to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> and <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$"> respectively.
+ Let <!-- MATH
+ $e_+ =F_+(\hat{e}_i)$
+ -->
+<IMG
+ WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img286.gif"
+ ALT="$ e_+ =F_+(\hat{e}_i)$"> and <!-- MATH
+ $e_- =F_-(\hat{e}_j)$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img287.gif"
+ ALT="$ e_- =F_-(\hat{e}_j)$">.
+ We write <!-- MATH
+ $\underline t_+$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img288.gif"
+ ALT="$ \underline t_+ $"> for the tangential unit vector to <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">, oriented
+ counterclockwise with respect to <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> and <!-- MATH
+ $\underline t_- = -\underline t_+$
+ -->
+<IMG
+ WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img289.gif"
+ ALT="$ \underline t_- = -\underline t_+$">
+ for the respective from <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$">. For line integrals over the edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> we write <!-- MATH
+ $\int_{e_+}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img290.gif"
+ ALT="$ \int_{e_+}$"> if we chose the orientation induced by
+ <!-- MATH
+ $\underline t_+$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img288.gif"
+ ALT="$ \underline t_+ $"> and <!-- MATH
+ $\int_{e_-}$
+ -->
+<IMG
+ WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img291.gif"
+ ALT="$ \int_{e_-}$"> for the orientation of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> induced by <!-- MATH
+ $\underline t_-$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img292.gif"
+ ALT="$ \underline t_-$">.
+<BR>
+In order to obtain an <!-- MATH
+ $H(\mathop{\rm curl}; \Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method, proposition <A HREF="node1.html#prop:_no_jumps">1</A> tells us that we must ensure the continuity of the
+ tangential components of the global shape functions, that is
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_conformity_of_shape_func"></A><!-- MATH
+ \begin{equation}
+\underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="182" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img293.gif"
+ ALT="$\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(26)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ The following lemma will justify the choice of the moments describing the edge dofs. A consequence of the lemma will be,
+ that the matching of the local edge dofs <!-- MATH
+ $\alpha^{[K_+]}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img294.gif"
+ ALT="$ \alpha^{[K_+]}$"> and <!-- MATH
+ $\alpha^{[K_-]}$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img295.gif"
+ ALT="$ \alpha^{[K_-]}$"> guarantees
+ the pointwise condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
+ </FONT><P>
+<DIV><A NAME="lemma:_trace"><B>L<SMALL>EMMA</SMALL> 2</B></A>
+Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> denote the reference triangle and <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> one of its edges, parametrized by <!-- MATH
+ $\hat{e} \ni \hat{x}(s) := \underline a +
+s\, \hat{\underline t}$
+ -->
+<IMG
+ WIDTH="143" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img296.gif"
+ ALT="$ \hat{e} \ni \hat{x}(s) := \underline a +
+s\, \hat{\underline t}$">.
+ Let <!-- MATH
+ $\hat{\underline p} \in \mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img297.gif"
+ ALT="$ \hat{\underline p} \in \mathcal{S}^k$">, <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$"> as defined in (<A HREF="node2.html#def:_space_Sk">4</A>). It then holds
+ <!-- MATH
+ \begin{displaymath}
+(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="146" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img298.gif"
+ ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
+$">
+</DIV><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ </FONT><!-- MATH
+ \begin{displaymath}
+\hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad \textrm{for } i=1,2,3:
+ \quad \hat{p}_i(\hat{x}) = \prod_{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
+ \textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="545" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img299.gif"
+ ALT="$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad ...
+...{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
+\textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+ Hence, with the parametrization of <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img127.gif"
+ ALT="$ \hat{e}$"> by <!-- MATH
+ $\hat{x}(s)$
+ -->
+<IMG
+ WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img300.gif"
+ ALT="$ \hat{x}(s)$">
+ </FONT><!-- MATH
+ \begin{displaymath}
+\hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
+ \hat{\varphi }_{k-1}(s)\,,
+
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="390" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
+ SRC="img301.gif"
+ ALT="$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
+\hat{\varphi }_{k-1}(s)\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ with <!-- MATH
+ $\hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$
+ -->
+<IMG
+ WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img302.gif"
+ ALT="$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$">,
+ and
+ </FONT><!-- MATH
+ \begin{displaymath}
+(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} = s^k\,\sum_{i=1}^3\hat{t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
+ \hat{\varphi }_{k-1}(s)\,.
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="317" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
+ SRC="img303.gif"
+ ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} = s^...
+...t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
+\hat{\varphi }_{k-1}(s)\,.
+$">
+</DIV><P></P><FONT SIZE="-1">
+ We observe that the coefficient of <IMG
+ WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img304.gif"
+ ALT="$ s^k$"> is exactly <!-- MATH
+ $\hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$
+ -->
+<IMG
+ WIDTH="54" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img305.gif"
+ ALT="$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$">. By the definition of the space
+ <!-- MATH
+ $\mathcal{S}^k$
+ -->
+<IMG
+ WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img107.gif"
+ ALT="$ \mathcal{S}^k$">, this expression must vanish.
+
+</FONT>
+<P>
+<P>
+<DIV><B>R<SMALL>EMARK</SMALL> 5</B>
+In the case of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> being a quadrilateral, we have <!-- MATH
+ $\hat{R} = \mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img306.gif"
+ ALT="$ \hat{R} = \mathcal{P}^k$">. By the definition of <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$"> we see
+ immediately that here also <!-- MATH
+ $(\hat{\underline v}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img307.gif"
+ ALT="$ (\hat{\underline v}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$">.
+ </DIV><P></P>
+
+<P>
+<FONT SIZE="-1">The next proposition tells us how exactly to define element shape functions on a mapped element <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> in order to get
+ <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions.
+ </FONT><P>
+<DIV><B>P<SMALL>ROPOSITION</SMALL> 8</B>
+Condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>) is satiesfied, if we define the element shape functions <!-- MATH
+ $\underline N_+$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img285.gif"
+ ALT="$ \underline N_+$"> and <!-- MATH
+ $\underline N_-$
+ -->
+<IMG
+ WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img284.gif"
+ ALT="$ \underline N_-$">
+ by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) and take into account the orientation of the edge <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$">:
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_trafo_shape_functions"></A><!-- MATH
+ \begin{equation}
+\underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_+^{-T} \hat{\underline N}_i \,,
+ \qquad \underline N_- := -\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.
+
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="508" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img308.gif"
+ ALT="$\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_...
+...\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(27)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P></DIV><P></P>
+
+<P>
+<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
+ Let <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> be the reference element and <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> its affine or bilinear image.
+ Let <!-- MATH
+ $\underline v := \mathcal{P}_K(\hat{\underline v})$
+ -->
+<IMG
+ WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img309.gif"
+ ALT="$ \underline v := \mathcal{P}_K(\hat{\underline v})$"> be a vector field on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">, defined by the Piola transformation of a reference vector field
+ <!-- MATH
+ $\hat{\underline v} \in \hat{R}$
+ -->
+<IMG
+ WIDTH="50" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img310.gif"
+ ALT="$ \hat{\underline v} \in \hat{R}$">. Let <IMG
+ WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img66.gif"
+ ALT="$ e$"> be one of the edges of <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> and <!-- MATH
+ $\underline t$
+ -->
+<IMG
+ WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img12.gif"
+ ALT="$ \underline t$">
+ the tangent according to convention <A HREF="node1.html#def:_tangent">1</A>.
+<BR>
+In the case of <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> being a triangle, we have <!-- MATH
+ $\hat{R} = \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img91.gif"
+ ALT="$ \hat{R} = \mathcal{R}^k$">.
+ By the definition of the space <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$">, lemma <A HREF="node2.html#lemma:_trace">2</A> and <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> we can
+ conclude that <!-- MATH
+ $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img311.gif"
+ ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
+<BR>
+If <IMG
+ WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
+ SRC="img81.gif"
+ ALT="$ \hat{K}$"> is a quadrilateral, the previous remark and
+ <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> also tell us that <!-- MATH
+ $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
+ -->
+<IMG
+ WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img311.gif"
+ ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
+<BR>
+Hence the condition
+ </FONT><!-- MATH
+ \begin{displaymath}
+\int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underline N_-\cdot \underline t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="401" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
+ SRC="img312.gif"
+ ALT="$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underli...
+...ne t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
+$">
+</DIV><P></P><FONT SIZE="-1">
+ on the edge moments
+ is sufficient for the global edge shape functions to satiesfy (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
+ Note that
+<BR><!-- MATH
+ $\int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$
+ -->
+<IMG
+ WIDTH="307" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img313.gif"
+ ALT="$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$">.
+ So, by the definition (<A HREF="node2.html#eq:_trafo_shape_functions">27</A>) of the element shape functions on <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img57.gif"
+ ALT="$ K_+$"> resp. on <IMG
+ WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.gif"
+ ALT="$ K_-$">,
+ by the invariance of the dofs (proposition <A HREF="node2.html#prop:_Dof_invariance">7</A>) and by the definition of the
+ reference shape functions (example <A HREF="node2.html#ex:_shape_functions">3</A>) we have
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><!-- MATH
+ \begin{equation*}
+\int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \int_{\hat{e}_i} (\hat{\underline N}_i\cdot \hat{\underline t}_i)\hat{\varphi }\,d\hat{s} = 1
+ \qquad \textrm{and} \qquad
+ \int_{e_-} (\underline N_- \cdot \underline t_-)\varphi \,ds = -\int_{\hat{e}_j} (\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.
+\end{equation*}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="740" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
+ SRC="img314.gif"
+ ALT="$\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \i...
+...\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+ </TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+
+</FONT>
+<P>
+<FONT SIZE="-1">To close this section, let us make a note on the interpretation of the dofs on an element <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> in the case of lowest order polynomial
+ degree. In this case, all dofs are edge dofs, the degrees of freedom are <!-- MATH
+ $\hat{\alpha}_j(\hat{\underline
+v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$
+ -->
+<IMG
+ WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img315.gif"
+ ALT="$ \hat{\alpha}_j(\hat{\underline
+v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$"> and the tangential traces of shape functions are constant on each edge.
+ Since we require <!-- MATH
+ $\hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$
+ -->
+<IMG
+ WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img316.gif"
+ ALT="$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$"> for the reference shape functions, we have
+ </FONT><!-- MATH
+ \begin{displaymath}
+v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j \cdot \hat{\underline t})\,|\hat{e}_j| = ({\underline N}_j \cdot {\underline t}_j)\, |e_j|\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="313" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img317.gif"
+ ALT="$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j ...
+...\hat{e}_j\vert = ({\underline N}_j \cdot {\underline t}_j)\, \vert e_j\vert\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ where for the last equality we have used lemma <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A>. We see that
+ the dof <!-- MATH
+ $\alpha_j(\underline v)$
+ -->
+<IMG
+ WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img318.gif"
+ ALT="$ \alpha_j(\underline v)$"> 'sitting' on the edge <IMG
+ WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img319.gif"
+ ALT="$ e_j$"> is the value of the <I>scaled</I> tangential component
+ <!-- MATH
+ $|e_j|\left(\underline v\cdot \underline t_j\right)|_e$
+ -->
+<IMG
+ WIDTH="100" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img320.gif"
+ ALT="$ \vert e_j\vert\left(\underline v\cdot \underline t_j\right)\vert _e$">.
+<BR> </FONT><P>
+<DIV><B>R<SMALL>EMARK</SMALL> 6</B>
+For the invariance of the edge dofs it is essential that the moments <!-- MATH
+ $\alpha^{[K]}$
+ -->
+<IMG
+ WIDTH="38" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="img321.gif"
+ ALT="$ \alpha^{[K]}$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$"> are defined by using the
+ <I>unit</I> tangent
+ vector <!-- MATH
+ $\underline t = \frac{|\hat{e}|}{|e|}\,(\hat{D}F)\,\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="108" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
+ SRC="img322.gif"
+ ALT="$ \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert}\,(\hat{D}F)\,\hat{\underline t}$"> on <IMG
+ WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img86.gif"
+ ALT="$ K$">. If not, e. g. if we
+ just used the tangent <!-- MATH
+ $\tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$
+ -->
+<IMG
+ WIDTH="87" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img323.gif"
+ ALT="$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$">, we would lose the invariance of the dofs. In that case the dofs would
+ scale by a factor depending on the size of the edge or face ([<A
+ HREF="node4.html#Ned1">8</A>], remark on p. 326).
+<BR> </DIV><P></P>
+
+<P>
+<FONT SIZE="-1"></FONT>
+<H2><A NAME="SECTION00025000000000000000">
+2.5 Approximation and convergence results</A>
+</H2><FONT SIZE="-1">
+ Without going into details, we will cite here some results on approximation properties and convergence of Nédélec FEM of first type.
+
+<BR>
+We are in the setting of a
+ <I>conforming</I> FEM and have quasi-optimal approximation properties of the FE-spaces <!-- MATH
+ $V_h \subset H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img324.gif"
+ ALT="$ V_h \subset H(\mathop{\rm curl};\Omega )$">
+ </FONT><!-- MATH
+ \begin{displaymath}
+\| \underline u - \Pi_h^k \underline u\|_{H(\mathop{\rm curl}; \Omega )} = C\,\inf_{w\in V_h}\| \underline u - \underline w\|_{H(\mathop{\rm curl}; \Omega )}\,,
+\end{displaymath}
+ -->
+<P></P>
+<DIV ALIGN="CENTER">
+<IMG
+ WIDTH="355" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img325.gif"
+ ALT="$\displaystyle \Vert \underline u - \Pi_h^k \underline u\Vert _{H(\mathop{\rm cu...
+...V_h}\Vert \underline u - \underline w\Vert _{H(\mathop{\rm curl}; \Omega )}\,,
+$">
+</DIV><P></P><FONT SIZE="-1">
+ where <!-- MATH
+ $\Pi_h^k \underline u \in \mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="82" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img326.gif"
+ ALT="$ \Pi_h^k \underline u \in \mathcal{R}^k$"> or <!-- MATH
+ $\Pi_h^k \underline u \in \mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img327.gif"
+ ALT="$ \Pi_h^k \underline u \in \mathcal{P}^k$"> respectively,
+ denotes the interpolate of <!-- MATH
+ $\underline u$
+ -->
+<IMG
+ WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img35.gif"
+ ALT="$ \underline u$"> with regard to the Nédélec dofs: <!-- MATH
+ $\alpha(\underline u) = \alpha(\Pi_h^k \underline u)$
+ -->
+<IMG
+ WIDTH="121" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img328.gif"
+ ALT="$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$"> for all dofs <IMG
+ WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img329.gif"
+ ALT="$ \alpha$">.
+ The interpolation operator <IMG
+ WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img330.gif"
+ ALT="$ \Pi_h^k$"> is defined for sufficiently smooth vector fields, namely for all <!-- MATH
+ $\underline v\in H^r(\mathop{\rm curl})$
+ -->
+<IMG
+ WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img331.gif"
+ ALT="$ \underline v\in H^r(\mathop{\rm curl})$">
+ for any <!-- MATH
+ $r>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img332.gif"
+ ALT="$ r>\frac{1}{2}$"> (see [<A
+ HREF="node4.html#Alonso-Valli">1</A>], Lemma 5.1., [<A
+ HREF="node4.html#Monk">7</A>] and references therein).
+<BR>
+For Nédélec's FEM of first type we state (without proof) the following optimal estimate in the curl-norm:
+ </FONT><P>
+<DIV><B>T<SMALL>HEOREM</SMALL> 5</B>
+If <!-- MATH
+ $\mathcal{T}_h$
+ -->
+<IMG
+ WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img333.gif"
+ ALT="$ \mathcal{T}_h$">, <IMG
+ WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img334.gif"
+ ALT="$ h>0$">, is a regular family of triangulations on <IMG
+ WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img10.gif"
+ ALT="$ \Omega $"> and <!-- MATH
+ $r>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img332.gif"
+ ALT="$ r>\frac{1}{2}$">, then there exists a constant <IMG
+ WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="img335.gif"
+ ALT="$ C>0$">,
+ depending on <IMG
+ WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img336.gif"
+ ALT="$ r$"> but not on <IMG
+ WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img337.gif"
+ ALT="$ h$"> or <!-- MATH
+ $\underline v$
+ -->
+<IMG
+ WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img20.gif"
+ ALT="$ \underline v$">, such that
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_curl-convergence"></A><!-- MATH
+ \begin{equation}
+\| \underline v - \Pi_h^k \underline v\|_{H(\mathop{\rm curl}; \Omega )} \leq C\,h^{\min\{r,k\}} \|\underline v\|_{H^r(\mathop{\rm curl};\Omega )}\,,
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="348" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
+ SRC="img338.gif"
+ ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{H(\mathop{\rm cu...
+... C\,h^{\min\{r,k\}} \Vert\underline v\Vert _{H^r(\mathop{\rm curl};\Omega )}\,,$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(28)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+for all <!-- MATH
+ $\underline v\in H^r(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img339.gif"
+ ALT="$ \underline v\in H^r(\mathop{\rm curl};\Omega )$">.
+ </DIV><P></P>
+<FONT SIZE="-1">
+ The result in (<A HREF="node2.html#eq:_curl-convergence">28</A>) was obtained by Alonso and Valli in [<A
+ HREF="node4.html#Alonso-Valli">1</A>], extending earlier interpolation
+ results by Nédélec in [<A
+ HREF="node4.html#Ned1">8</A>] and Monk in [<A
+ HREF="node4.html#Monk_92">6</A>].
+<BR>
+Optimal convergence in the <!-- MATH
+ $H(\mathop{\rm curl};\Omega )$
+ -->
+<IMG
+ WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img1.gif"
+ ALT="$ H(\mathop {\rm curl};\Omega )$">-norm for the error of the FE-approximation of the model
+ problem (<A HREF="node1.html#eq:_varform">3</A>) by Nédélec's elements of first type follows from (<A HREF="node2.html#eq:_curl-convergence">28</A>) by Céa's lemma.
+ This result has been verified in numerical experiments with a <TT>MATLAB</TT> code, which uses lowest
+ order Nédélec elements on affine triangular meshes for 2d problems, as well as with a <TT>deal.II</TT> code, which uses lowest order
+ Nédélec elements on bilinear resp. trilinear meshes for 2d resp. 3d problems.
+<BR>
+As for the <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-approximation properties of FE spaces based on <!-- MATH
+ $\mathcal{R}^k$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img102.gif"
+ ALT="$ \mathcal{R}^k$"> or <!-- MATH
+ $\mathcal{P}^k$
+ -->
+<IMG
+ WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="img228.gif"
+ ALT="$ \mathcal{P}^k$">, we could hope for a better order
+ than <!-- MATH
+ $\mathcal{O}(h^k)$
+ -->
+<IMG
+ WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img340.gif"
+ ALT="$ \mathcal{O}(h^k)$"> at first sight: still, we have <!-- MATH
+ $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$
+ -->
+<IMG
+ WIDTH="165" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img341.gif"
+ ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$">. However, Nédélec shows in [<A
+ HREF="node4.html#Ned1">8</A>]
+ that only suboptimality can be expected:
+ </FONT><P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_L2-approx"></A><!-- MATH
+ \begin{equation}
+\| \underline v - \Pi_h^k \underline v\|_{L^2(\Omega )} \leq C h^k |\underline v|_{H^k(\Omega )}\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="244" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
+ SRC="img342.gif"
+ ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{L^2(\Omega )} \leq C h^k \vert\underline v\vert _{H^k(\Omega )}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(29)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
+ Nédélec uses a standard scaling and Bramble-Hilbert argument to derive (<A HREF="node2.html#eq:_L2-approx">29</A>). Since
+ <!-- MATH
+ $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$
+ -->
+<IMG
+ WIDTH="254" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img343.gif"
+ ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$">, the Bramble-Hilbert argument only guarantees an elementwise
+ approximation of order <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> of <IMG
+ WIDTH="60" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img344.gif"
+ ALT="$ H^k(K)$">-functions from the space <!-- MATH
+ $\mathcal{R}^k(K)$
+ -->
+<IMG
+ WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img345.gif"
+ ALT="$ \mathcal{R}^k(K)$">.
+<BR>
+However, in a recent paper Hiptmair uses a duality technique to state optimal convergence of the
+ <!-- MATH
+ $L^2(\Omega )$
+ -->
+<IMG
+ WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ L^2(\Omega )$">-error <!-- MATH
+ $\| \underline u - \underline u_h\|_{L^2(\Omega )}$
+ -->
+<IMG
+ WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img346.gif"
+ ALT="$ \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )}$">
+ for the 3d case and Nédélec's elements of first type of order <IMG
+ WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img93.gif"
+ ALT="$ k$"> on tetrahedral meshes
+ (see Section 5.3, Theorem 5.8 in [<A
+ HREF="node4.html#Hipt">5</A>]):
+ </FONT><P>
+<DIV><A NAME="th:_Hiptmair"><B>T<SMALL>HEOREM</SMALL> 6</B></A>
+There is an <!-- MATH
+ $s>\frac{1}{2}$
+ -->
+<IMG
+ WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img347.gif"
+ ALT="$ s>\frac{1}{2}$"> such that
+ <P></P>
+<DIV ALIGN="CENTER"><A NAME="eq:_L2-convergence"></A><!-- MATH
+ \begin{equation}
+\| \underline u - \underline u_h\|_{L^2(\Omega )} \leq C h^s \| \underline u - \underline u_h\|_{H(\mathop{\rm curl}; \Omega )}\,.
+\end{equation}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><IMG
+ WIDTH="299" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="img348.gif"
+ ALT="$\displaystyle \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )} \leq C h^s \Vert \underline u - \underline u_h\Vert _{H(\mathop{\rm curl}; \Omega )}\,.$"></TD>
+<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
+(30)</TD></TR>
+</TABLE></DIV>
+<BR CLEAR="ALL"><P></P>
+Under the assumption that the boundary <!-- MATH
+ $\partial \Omega$
+ -->
+<IMG
+ WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img9.gif"
+ ALT="$ \partial \Omega $"> is smooth or convex, <IMG
+ WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ SRC="img349.gif"
+ ALT="$ s=1$"> can be chosen.
+ </DIV><P></P>
+<FONT SIZE="-1">
+ Several key arguments of the proof in [<A
+ HREF="node4.html#Hipt">5</A>] make explicitely use of features that are limited to 3d problems and the
+ family of finite elements based on tetrahedrons.
+ They cannot be modified trivially to apply to 2d problems or 3d problems on hexahedral meshes. Even worse, it is suggested by the results of
+ numerical experiments that one cannot hope to obtain a result similar to (<A HREF="node2.html#eq:_L2-convergence">30</A>).
+<BR>
+A possibility to overcome this deficiency of convergence is to use Nédélec elements of second type, where the full <!-- MATH
+ $[\mathbb{P}_k]^d$
+ -->
+<IMG
+ WIDTH="43" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="img350.gif"
+ ALT="$ [\mathbb{P}_k]^d$">
+ are used as polynomial spaces (see [<A
+ HREF="node4.html#Ned2">10</A>]).
+</FONT>
+<P>
+<FONT SIZE="-1"></FONT>
+<BR><HR>
+<ADDRESS>
+
+2003-04-30
+</ADDRESS>
+</BODY>
+</HTML>
projects / dealii-svn.git / commitdiff
