From: guido Date: Tue, 16 Nov 2004 13:32:50 +0000 (+0000) Subject: make PDF version default ad delete HTML X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=58b37c0b4af122dbd61f9007ca6d23687f60193f;p=dealii-svn.git make PDF version default ad delete HTML git-svn-id: https://svn.dealii.org/trunk@9769 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/doc/reports/nedelec/images.aux b/deal.II/doc/reports/nedelec/images.aux deleted file mode 100644 index c4cb9640ce..0000000000 --- a/deal.II/doc/reports/nedelec/images.aux +++ /dev/null @@ -1,2 +0,0 @@ -\relax -\bibstyle{abbrv} diff --git a/deal.II/doc/reports/nedelec/images.bbl b/deal.II/doc/reports/nedelec/images.bbl deleted file mode 100644 index 581abba97b..0000000000 --- a/deal.II/doc/reports/nedelec/images.bbl +++ /dev/null @@ -1,78 +0,0 @@ -\begin{thebibliography}{10} - -\bibitem{Alonso-Valli} -A.~Alonso and A.~Valli. -\newblock An optimal domain decomposition preconditioner for low-frequency - time-harmonic {M}axwell equations. -\newblock {\em Math.~Comp.}, 68(226):607--631, 1999. - -\bibitem{Deal} -W.~Bangerth, R.~Hartmann, and G.~Kanschat. -\newblock {\em {\tt deal.{I}{I}} Differential Equations Analysis Library, - Technical Reference}. -\newblock IWR, Universit{\"a}t Heidelberg. -\newblock \texttt{http://www.dealii.org}. - -\bibitem{Brezzi-Fortin} -F.~Brezzi and M.~Fortin. -\newblock {\em Mixed and Hybrid Finite Element Methods}, volume~15 of {\em - Springer Series in Computational Mathematics}. -\newblock Springer-Verlag, New York, 1991. - -\bibitem{Girault-Raviart} -V.~Girault and P.-A. Raviart. -\newblock {\em Finite Element Approximation of the Navier-Stokes Equations}, - volume 749 of {\em Lecture Notes in Mathematics}. -\newblock Springer-Verlag, Berlin, Heidelberg, 1979, 1981. - -\bibitem{Hipt} -R.~Hiptmair. -\newblock Finite elements in computational electromagnetism. -\newblock In {\em Acta Numerica}, pages 1--103. {C}ambridge {U}niversity press, - 2002. - -\bibitem{Monk'92} -P.~Monk. -\newblock Analysis of a finite element method for {M}axwell's equations. -\newblock {\em SIAM J.~Numer.~Anal}, 29:714--729, 1992. - -\bibitem{Monk} -P.~Monk. -\newblock A simple proof for an edge element discretization of {M}axwell's - equations. -\newblock Submitted for publication. Download version available on Monk's - webpage: www.math.udel.edu./~monk, 2001. - -\bibitem{Ned1} -J.~C. N\'ed\'elec. -\newblock Mixed finite elements in $\mathbb{R}^3$. -\newblock {\em Numer.~Math.}, 35:315--341, 1980. - -\bibitem{Ned3} -J.~C. N\'ed\'elec. -\newblock Elements finis mixtes incompressibles pour l'\'equation de {S}tokes - dans $\mathbb{R}^3$. -\newblock {\em Numer.~Math.}, 39:97--112, 1982. - -\bibitem{Ned2} -J.~C. N\'ed\'elec. -\newblock A new family of mixed finite elements in $\mathbb{R}^3$. -\newblock {\em Numer.~Math.}, 50:57--81, 1986. - -\bibitem{Demko} -W.~Rachowicz and L.~Demkowicz. -\newblock A two-dimensional hp-adaptive finite element package for - electromagnetics (2{D}hp90\_{E}{M}). -\newblock Ticam Report 98--16, TICAM, 1998. -\newblock Download version available on Demkowicz' webpage: - www.ticam.utexas.edu/~{L}eszek. - -\bibitem{Demko3d} -W.~Rachowicz and L.~Demkowicz. -\newblock A three-dimensional hp-adaptive finite element package for - electromagnetics (3{D}hp90\_{E}{M}). -\newblock Ticam Report 00-04.2000, TICAM, 2000. -\newblock Download version available on Demkowicz' webpage: - www.ticam.utexas.edu/~{L}eszek. - -\end{thebibliography} diff --git a/deal.II/doc/reports/nedelec/images.log b/deal.II/doc/reports/nedelec/images.log deleted file mode 100644 index d036f77a48..0000000000 --- a/deal.II/doc/reports/nedelec/images.log +++ /dev/null @@ -1,2433 +0,0 @@ -This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2002.9.6) 30 APR 2003 17:04 -**./images.tex -(./images.tex -LaTeX2e <2000/06/01> -Babel and hyphenation patterns for american, french, german, ngerman, i -talian, nohyphenation, loaded. - 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- -] -! 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Missing } inserted. - - } -l.2552 ...ad \partial \Omega \,, ^^I^^I\end{split} - \end{displaymath}% -I've put in what seems to be necessary to fix -the current column of the current alignment. -Try to go on, since this might almost work. - -l2hSize :displaymath7133:34.629pt::0.0pt::349.0pt. -[374 - - -] -l2hSize :tex2html_wrap_inline7135:10.02922pt::10.02922pt::48.12807pt. -[375 - - -] -l2hSize :tex2html_wrap_inline7137:16.83748pt::16.83748pt::134.97365pt. -[376 - - -] -l2hSize :tex2html_wrap_indisplay7141:10.48782pt::10.48782pt::187.09563pt. -[377 - - -] -l2hSize :tex2html_wrap_inline7143:7.98248pt::7.98248pt::9.2619pt. -[378 - - -] -l2hSize :tex2html_wrap_inline7147:9.52922pt::0.0pt::62.02588pt. -[379 - - -] -! 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\underline u)_{H(\mathop{\rm curl};\Omega )} := (\underline v, \underline u)_{L^2(\Omega )} + (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )} -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5539}% -$ [\mathcal{D}(\overline{\Omega })]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5546}% -$ \underline u$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5548}% -$ [H(\mathop{\rm curl};\Omega )]^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5552}% -$ H^1(\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5554}% -$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx = \int_{\Omega } \underline u \cdot \mathop{\underline{\rm curl}}\varphi \,dx + \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5558}% -$ [H(\mathop{\rm curl};\Omega )]^3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5562}% -$ [H^1(\Omega )]^3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5564}% -$\displaystyle \int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \,dx = \int_{\Omega } \underline u \cdot \mathop{\rm curl}\underline v \,dx + \int_{\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5566}% -$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5568}% -$\displaystyle \mathrm{div}\, (\underline u \wedge \underline v) = \underline v \cdot \mathop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5570}% -$ (\underline a\wedge\underline b)\cdot \underline c$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5572}% -$\displaystyle \int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - \underline u\cdot \mathop{\rm curl}\underline v \, dx = \int_{\Omega } \mathrm{div}\, (\underline u \wedge \underline v) \, dx -= \int_{\partial \Omega } (\underline u \wedge \underline v)\cdot \underline n \,ds = \int_{\partial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5574}% -$ H(\mathop{\rm curl})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5581}% -$ \underline n$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5591}% -$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5595}% -$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5599}% -$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5603}% -$ [H^1(\Omega )]^{\tilde{d}}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5605}% -$ [H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5610}% -$\displaystyle H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mathop{\rm curl};\Omega ): \quad \underline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\} -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5615}% -$ [\mathcal{D}(\Omega )]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5617}% -$ H_0(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5622}% -$ K_-$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5624}% -$ K_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5626}% -$ \mathbb{R}^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5628}% -$ e = \partial K_-\cap\partial K_+ \neq \emptyset$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5630}% -$ \Omega = \partial K_-\cup\partial K_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5632}% -$ v$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5636}% -$ v_-$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5642}% -$ H(\mathop{\rm curl}; K_-)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5644}% -$ v_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5650}% -$ H(\mathop{\rm curl}; K_+)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5652}% -$ e$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5654}% -$ v_-\wedge n_- + v_+\wedge n_+ = 0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5658}% -$ H^{\frac{1}{2}}_{00}(e)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5662}% -$ H^{\frac{1}{2}}(\partial \Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5674}% -$ H^{-\frac{1}{2}}(e)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5676}% -$ \underline v_-$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5678}% -$ \underline v_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5680}% -$ L^2(e)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5684}% -$ H^1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5694}% -$ \underline u \in H_0(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5696}% -$ \underline v \in H_0(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5698}% -$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx = \int_{\Omega } \underline f\, \cdot \underline v\,dx$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath5701}% -\begin{displaymath}\begin{split} a(\underline u,\underline v) &:= \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx \\ l(\underline v) & := \int_{\Omega } \underline f\, \cdot \underline v\,dx \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5703}% -$ a(\cdot,\cdot)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5705}% -$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{section} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5717}% -$ \hat{K}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5719}% -$ F_K(\hat{x})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5721}% -$ K = F_K(\hat{K})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5723}% -$ \hat{R}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5727}% -$ R_K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5729}% -$ K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5731}% -$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5735}% -$ N < \infty$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5739}% -$ \mathcal{A}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5741}% -$ \alpha_i(\cdot)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5749}% -$ \hat{R} = \mathcal{R}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5751}% -$ \mathbb{P}_k(\hat{\Sigma})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5753}% -$ k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5755}% -$ \hat{\Sigma}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5759}% -$ \tilde{\mathbb{P}}_k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5765}% -$ d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5770}% -$\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d : \underline p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5772}% -$ \hat{x} \in \hat{K}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5778}% -$ k(k+2)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5785}% -$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath5787}% -\begin{displaymath}\begin{split} \mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \textrm{for} \quad d=2\,, \\ \mathrm{dim} (\mathcal{R}^k) &= \frac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5789}% -$ \mathcal{R}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5791}% -$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5795}% -$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right) \subseteq \mathcal{S}^k -%%\left\{\v p\quad \big|\quad \v p = \tilde{p} \left(\begin{array}{cc} x_2 \\-x_1 \end{array}\right)\,,\, \tilde{p} - $% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5797}% -$ \tilde{\mathbb{P}}_{k-1}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5799}% -$ k-1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5803}% -$ \mathcal{S}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5809}% -$ k=1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5811}% -$ k=2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5814}% -$\displaystyle \mathcal{R}^1 = \left\langle \left(\begin{array}{cc} 1 \\ 0 \end{array}\right)\,_, \left(\begin{array}{cc} 0 \\ 1 \end{array}\right)\,_, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right) \right\rangle$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5816}% -$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus -\left\langle -\left(\begin{array}{cc} \hat{x}_1\,\hat{x}_2 \\ -{\hat{x}_1}^2 \end{array}\right)\,_, -\left(\begin{array}{cc} {\hat{x}_2}^2 \\ -\hat{x}_1\,\hat{x}_2 \end{array}\right) -\right\rangle -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5823}% -$ \mathcal{S}^1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5825}% -$ \underline p$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5827}% -$ (\mathbb{P}_{1}(\hat{K}))^3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5829}% -$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5835}% -$\displaystyle \underline p \cdot \hat{\underline x} = -\sum_{i=1}^3 a_{ii}\hat{x}_i^2 + \sum_{\substack{i,j=1 \\ j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath5839}% -\begin{displaymath}\begin{split} &a_{11}=a_{22}=a_{33} = 0 \\ &a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,. \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5843}% -$ a_{ij} = 1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5845}% -$ i=1,2,3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5847}% -$ j>i$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5849}% -$\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus \left\langle \left(\begin{array}{ccc} 0 \\ \hat{x}_3 \\ \hat{x}_2 \end{array}\right)\,_,\, \left(\begin{array}{ccc} \hat{x}_3 \\ 0 \\ \hat{x}_1 \end{array}\right)\,_,\, \left(\begin{array}{ccc} \hat{x}_2 \\ \hat{x}_1 \\ 0 \end{array}\right) \right\rangle$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5853}% -$ (\mathbb{P}_{k}(\hat{K}) )^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5868}% -$ n$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5870}% -$ n+k+2 \choose n$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5875}% -$ \hat{\underline t}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5883}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5885}% -$ \hat{e}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5889}% -$ 3k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5891}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5893}% -$ k(k-1)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5900}% -$ \hat{\underline n}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5915}% -$ 6k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5917}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \quad \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5919}% -$ \hat{f}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5923}% -$ 4k(k-1)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5925}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5927}% -$ \frac{k(k-1)(k-2)}{2}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5933}% -$ k\leq3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5944}% -$ \hat{\underline u}\in \mathcal{R}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5946}% -$ \hat{\alpha}(\hat{\underline u})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5949}% -$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5951}% -$ \hat{e}_0 = \overline{(0,0),(1,0)}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5953}% -$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array}\right)\,,\quad -\hat{\underline t}_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} -1 \\ 1 \end{array}\right)\,,\quad -\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5955}% -$ \mathcal{R}^1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5961}% -$ \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5963}% -$ \varphi \equiv 1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5965}% -$ \mathbb{P}_{0}(\hat{e}_i)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5967}% -$\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5969}% -$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5973}% -$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5975}% -$ \hat{\underline N}_i$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5979}% -$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ \hat{x} \end{array}\right)\,,\quad \hat{\underline N}_1 = \left(\begin{array}{cc} -\hat{y} \\ \hat{x} \end{array}\right)\,,\quad \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\ \hat{x}-1 \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5984}% -$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5988}% -$ N_i$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline5992}% -$ \hat{N}_i$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay5996}% -$\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6000}% -$ H(\mathop{\rm curl};\hat{K})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6002}% -$ H(\mathop{\rm curl}; K)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6011}% -$ \underline N_i(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6015}% -$\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6017}% -$ \hat{D}F_K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6019}% -$ \frac{d}{d\hat{x}}F_K(\hat{x})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6022}% -$ H(\mathop{\rm div}; \Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6026}% -$ F_K(\hat{x}) = B_K \hat{x} + b_k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6030}% -$ B_K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6032}% -$\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6035}% -$ \Omega \subset\mathbb{R}^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6041}% -$\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6043}% -$ R$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6048}% -$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6050}% -$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6052}% -$ \hat{x} = -F_K^{-1}(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6054}% -$ F_K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6058}% -$ D\underline v$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6060}% -$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6062}% -$\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6074}% -$ R\,Dv$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6076}% -$ B_K^{-T}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6080}% -$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6085}% -$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6087}% -$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underline u \, dx = | B_K |^{-1}\,\int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6092}% -$ \hat{\underline v}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6096}% -$ \mathop{\rm Curl}v$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6098}% -$\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6100}% -$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6102}% -$\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6109}% -$ \underline v(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6113}% -$ \hat{\underline v}(\hat{x})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6115}% -$\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6117}% -$ \mathrm{M_i}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6119}% -$ D(F_K^{-1}) = B_K^{-1}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6121}% -$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6123}% -$\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll} (\widehat{\mathop{\rm curl}}\,\hat{v} \circ F_K^{-1})_k (x) & \textrm{if} \quad l=i \\ (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i \end{array} \right.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6125}% -$\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\mathop{\rm Curl}v})_{23} \\ ({\mathop{\rm Curl}v})_{31} \\ ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6127}% -$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6129}% -$ b_{ij} := (B_K^{-1})_{ij}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6131}% -$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3} -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6135}% -$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\widehat{\mathop{\rm Curl}} \,\hat{v})_{12} --(b_{12}b_{33} - b_{32}b_{13})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{31} -+(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6137}% -$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm curl}}\, v)_1 & b_{12} & b_{13} \\ -(\widehat{\mathop{\rm curl}}\, v)_2 & b_{22} & b_{23} \\ -(\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33} -\end{array}\right) \,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6148}% -$\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6150}% -$ (\mathop{\rm curl}\underline v)_1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6152}% -$\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6154}% -$ \det \mathrm{M_1}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6158}% -$\displaystyle \det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det\mathcal{B}^{inv}_{11} --(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det\mathcal{B}^{inv}_{21} -+(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6160}% -$ \mathcal{B}^{inv}_{ij}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6162}% -$ 2 \times 2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6164}% -$ B_K^{-1}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6166}% -$ A \in \mathbb{R}^{3\times 3}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6168}% -$\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6170}% -$ \mathcal{A}_{ij}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6174}% -$ A$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6178}% -$ A = B_K^{-1}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6180}% -$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \mathcal{B}^{inv}_{j1} (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_j -= (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det \mathcal{B}^{inv}_{11} --(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det \mathcal{B}^{inv}_{21} -+(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6183}% -$ C$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6185}% -$ \hat{C} = [0,1]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6187}% -$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6191}% -$ \mathcal{Q}_{l,m}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6193}% -$ \hat{C}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6195}% -$ l$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6197}% -$ \hat{x}_1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6199}% -$ m$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6201}% -$ \hat{x}_2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6203}% -$ \mathcal{Q}_{l,m,n}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6217}% -$ \hat{x}_3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6221}% -$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{cc} \hat{u}_1 \\ \hat{u}_2 \end{array}\right): \quad \hat{u}_1 \in \mathcal{Q}_{k-1,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6223}% -$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{ccc} \hat{u}_1 \\ \hat{u}_2 \\ \hat{u}_3 \end{array}\right):\quad \hat{u}_1 \in \mathcal{Q}_{k-1,k,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1,k}\,, \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6228}% -$ \hat{C}\subset \mathbb{R}^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6237}% -$ \mathcal{P}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6241}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6247}% -$ 4k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6249}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x}\,, \quad \forall\, \hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\ \hat{\varphi }_2 \end{array}\right) \,, \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6251}% -$ 2k(k-1)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6273}% -$ 12k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6275}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \,,\quad \forall \,\hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\ \hat{\varphi }_2 \end{array}\right) \,, \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6281}% -$ 6\cdot 2k(k-1)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6283}% -$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \,,\quad \forall\, \hat{\underline \varphi } = \left(\begin{array}{ccc} \hat{\varphi }_1 \\ \hat{\varphi }_2 \\ \hat{\varphi }_3\end{array}\right) \,,\quad\hat{\varphi }_1\in\mathcal{Q}_{k-1,k-2,k-2}\,,\quad\hat{\varphi _2}\in\mathcal{Q}_{k-2,k-1,k-2}\,, \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6285}% -$ 3k(k-1)^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6288}% -$ [0,1]^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6290}% -$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array}\right)\,,\quad -\hat{\underline t}_1 = \left(\begin{array}{cc} 0 \\ 1 \end{array}\right)\,,\quad -\hat{\underline t}_2 = \left(\begin{array}{cc} -1 \\ 0 \end{array}\right)\,, \quad -\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6292}% -$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ 0 \end{array}\right)\,,\quad \hat{\underline N}_1 = \left(\begin{array}{cc} 0 \\ \hat{x} \end{array}\right)\,,\quad \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\ 0 \end{array}\right)\,,\quad \hat{\underline N}_3 = \left(\begin{array}{cc} 0 \\ \hat{x}-1 \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6296}% -$ F_C(\hat{C})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6300}% -$ \hat{D}F_C(\hat{x})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6302}% -$ F_C$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6307}% -$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6315}% -$ \mathop{\rm curl}\underline v$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6317}% -$ \widehat{\mathop{\rm curl}}\,\hat{\underline v}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6330}% -$\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6334}% -$ F$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6338}% -$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6340}% -$ D(F^{-1})_{ij}(x)= -\frac{\partial \hat{x}_i}{\partial x_j}(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6342}% -$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath6344}% -\begin{displaymath}\begin{split} \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\ \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath6346}% -\begin{displaymath}\begin{split} \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) + \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\ \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) + \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6348}% -$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1} - \frac{\partial v_1}{\partial x_2} = -\frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) -- \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsubsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6353}% -$ D(F_C^{-1})(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6355}% -$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6357}% -$ i,j = 1,2,3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6359}% -$ \mathop{\rm Curl}v = Dv^T - Dv$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6361}% -$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,\hat{D}F_C^{-1}) \circ F_C^{-1})(x) -= (DF_C^{-1})^T(x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6363}% -$ B_C$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6365}% -$ \hat{D}F_C(\hat(x))$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6370}% -$ C = F_C(\hat{C})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6376}% -$\displaystyle \mathop{\rm curl}\underline v = \left(\frac{1}{\det \hat{D}F_C}\,\hat{D}F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6383}% -$ \mathcal{Q}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6389}% -$ \mathcal{P}_K$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6393}% -$ K=F(\hat{K})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6399}% -$ [0,|e|] \ni s \mapsto \underline x(s) \in e$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6401}% -$ [0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6411}% -$ \frac{d \underline x}{ds}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6413}% -$ \frac{d \hat{\underline x}}{d\hat{s}}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6422}% -$\displaystyle \underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6424}% -$ |\hat{e}|$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6426}% -$ |e|$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6432}% -$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6434}% -$ \hat{x}_j = \hat{x}_j(\underline x(s))$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6436}% -$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6438}% -$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \underline x}{ds} = \left( \hat{\underline v}_j\frac{\partial \hat{x}_j}{\partial x_i}\right) (x)\frac{dx_i}{ds} -= \hat{\underline v}_j \frac{d \hat{x}_j}{ds} = \hat{\underline v}_j \frac{d \hat{x}_j}{d\hat{s}} \frac{d\hat{s}}{ds} -= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds} -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6440}% -$ \frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6451}% -$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6453}% -$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline t)\varphi \,ds = -\int_{\hat{e}} (\hat{\underline v} \cdot \hat{\underline t}) \hat{\varphi } \, d\hat{s}\, = \hat{\alpha}(\hat{\underline u})\,, -\qquad \forall\, \hat{\varphi } \in -\mathbb{P}_{k-1}(\hat{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6455}% -$ K_- = F_-(\hat{K})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6457}% -$ K_+ = F_+(\hat{K})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6461}% -$ \underline N$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6465}% -$ \underline N_-$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6467}% -$ \underline N_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6475}% -$ e_+ =F_+(\hat{e}_i)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6477}% -$ e_- =F_-(\hat{e}_j)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6479}% -$ \underline t_+ $% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6485}% -$ \underline t_- = -\underline t_+$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6491}% -$ \int_{e_+}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6495}% -$ \int_{e_-}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6499}% -$ \underline t_-$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6503}% -$\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6505}% -$ \alpha^{[K_+]}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6507}% -$ \alpha^{[K_-]}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6514}% -$ \hat{e} \ni \hat{x}(s) := \underline a + -s\, \hat{\underline t}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6516}% -$ \hat{\underline p} \in \mathcal{S}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6520}% -$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6522}% -$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad \textrm{for } i=1,2,3: -\quad \hat{p}_i(\hat{x}) = \prod_{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad -\textrm{where } \sum_{j=1}^3 k_{ij} = k\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6526}% -$ \hat{x}(s)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6528}% -$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} + -\hat{\varphi }_{k-1}(s)\,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6530}% -$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6532}% -$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} = s^k\,\sum_{i=1}^3\hat{t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) + -\hat{\varphi }_{k-1}(s)\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6534}% -$ s^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6536}% -$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6543}% -$ \hat{R} = \mathcal{P}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6547}% -$ (\hat{\underline v}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6560}% -$\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_+^{-T} \hat{\underline N}_i \,, \qquad \underline N_- := -\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6566}% -$ \underline v := \mathcal{P}_K(\hat{\underline v})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6570}% -$ \hat{\underline v} \in \hat{R}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6584}% -$ (\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6590}% -$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underline N_-\cdot \underline t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e) -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6592}% -$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6598}% -$\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \int_{\hat{e}_i} (\hat{\underline N}_i\cdot \hat{\underline t}_i)\hat{\varphi }\,d\hat{s} = 1 \qquad \textrm{and} \qquad \int_{e_-} (\underline N_- \cdot \underline t_-)\varphi \,ds = -\int_{\hat{e}_j} (\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6602}% -$ \hat{\alpha}_j(\hat{\underline -v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6604}% -$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6606}% -$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j \cdot \hat{\underline t})\,|\hat{e}_j| = ({\underline N}_j \cdot {\underline t}_j)\, |e_j|\,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6608}% -$ \alpha_j(\underline v)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6610}% -$ e_j$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6612}% -$ |e_j|\left(\underline v\cdot \underline t_j\right)|_e$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6615}% -$ \alpha^{[K]}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6619}% -$ \underline t = \frac{|\hat{e}|}{|e|}\,(\hat{D}F)\,\hat{\underline t}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6623}% -$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{subsection} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6626}% -$ V_h \subset H(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6628}% -$\displaystyle \| \underline u - \Pi_h^k \underline u\|_{H(\mathop{\rm curl}; \Omega )} = C\,\inf_{w\in V_h}\| \underline u - \underline w\|_{H(\mathop{\rm curl}; \Omega )}\,, -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6630}% -$ \Pi_h^k \underline u \in \mathcal{R}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6632}% -$ \Pi_h^k \underline u \in \mathcal{P}^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6636}% -$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6638}% -$ \alpha$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6640}% -$ \Pi_h^k$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6642}% -$ \underline v\in H^r(\mathop{\rm curl})$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6644}% -$ r>\frac{1}{2}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6647}% -$ \mathcal{T}_h$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6649}% -$ h>0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6655}% -$ C>0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6657}% -$ r$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6659}% -$ h$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6663}% -$\displaystyle \| \underline v - \Pi_h^k \underline v\|_{H(\mathop{\rm curl}; \Omega )} \leq C\,h^{\min\{r,k\}} \|\underline v\|_{H^r(\mathop{\rm curl};\Omega )}\,,$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6665}% -$ \underline v\in H^r(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6675}% -$ \mathcal{O}(h^k)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6677}% -$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6679}% -$\displaystyle \| \underline v - \Pi_h^k \underline v\|_{L^2(\Omega )} \leq C h^k |\underline v|_{H^k(\Omega )}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6681}% -$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6685}% -$ H^k(K)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6687}% -$ \mathcal{R}^k(K)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6691}% -$ \| \underline u - \underline u_h\|_{L^2(\Omega )}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6696}% -$ s>\frac{1}{2}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6698}% -$\displaystyle \| \underline u - \underline u_h\|_{L^2(\Omega )} \leq C h^s \| \underline u - \underline u_h\|_{H(\mathop{\rm curl}; \Omega )}\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6702}% -$ s=1$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6704}% -$ [\mathbb{P}_k]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\stepcounter{section} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6711}% -$ \Omega = [-1,1]^d$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6715}% -$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\ 3 - x^2 \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6717}% -$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6721}% -$ 2^5$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6723}% -$ 2^{13}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6731}% -$ \mathcal{O}(h)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6733}% -$ L^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6735}% -$ \#$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6818}% -$ H(\mathop{\rm curl};\Omega )$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlpictureA{tex2html_wrap4265}% -% latex2html id marker 4265 -\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}% -\lthtmlpictureZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline6830}% -$ H(\mathop{\rm curl};(\Omega ))$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlpictureA{tex2html_wrap4271}% -% latex2html id marker 4271 -\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}% -\lthtmlpictureZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlpictureA{tex2html_wrap4293}% -\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}% -\lthtmlpictureZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay6913}% -$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{ccc} xy(1 - y^2)(1-z^2) + 2xy(1-z^2) \\ y^2(1 - x^2)(1-z^2) + (1-y^2)(2-x^2-z^2) \\ yz(1 - x^2)(1-y^2) + 2yz(1-x^2) \end{array}\right)\,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlpictureA{tex2html_wrap4329}% -\includegraphics[width=9.5cm, height=7cm]{field1.eps}% -\lthtmlpictureZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlpictureA{tex2html_wrap4335}% -\includegraphics[width=9.5cm, height=7cm]{field2.eps}% -\lthtmlpictureZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7106}% -$ \mathbb{R}^3$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\appendix -\stepcounter{section} -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7119}% -$ \mathbb{R}^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7121}% -$ \varphi (x,y)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7125}% -$ c>0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7127}% -$ w$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath7129}% -\begin{displaymath}\begin{split} -\Delta w + c\, w &= \varphi \quad \mathrm{in} \quad \Omega \\ \underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7131}% -$ \underline E := \nabla^{\perp} w$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath7133}% -\begin{displaymath}\begin{split} \underline \mathop{\rm curl}\mathop{\rm curl}\underline E + c\, \underline E = \underline f \quad \mathrm{in} \quad \Omega \,, \\ \underline E \wedge \underline n = 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7135}% -$ \underline f := \nabla^{\perp} \varphi $% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7137}% -$ \nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\ -\partial _x\varphi -\end{array}\right)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7141}% -$\displaystyle \underline E \wedge \underline n = \underline E \cdot \underline t = {\nabla w}^T \boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7143}% -$ \underline E$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7147}% -$ \nabla\cdot\nabla^{\perp}w = 0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7151}% -$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7153}% -$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7155}% -$ \nabla^{\perp}w$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7157}% -$ \nabla^{\perp} \varphi $% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath7162}% -\begin{displaymath}\begin{split} -\Delta w &= \lambda \, w \quad \mathrm{in} \quad \Omega \\ \underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7164}% -$ \varphi = (\lambda + c)\,w$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7166}% -$ \Omega = [-1,1]^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7168}% -$ \lambda = 2\pi^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7170}% -$ w = \cos\pi x\cos\pi y$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7172}% -$\displaystyle \underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,, \qquad -\underline E = \pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7179}% -$ w(x,y) = (1-x^2)^2(1-y^2)^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7181}% -$ \underline n \cdot \nabla w = 0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7183}% -$ \partial [-1,1]^2$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7185}% -$ \varphi = -\Delta w + c w$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -\appendix -\stepcounter{section} -{\newpage\clearpage -\lthtmldisplayA{displaymath7188}% -\begin{displaymath}\begin{split} \varepsilon \frac{\partial \mathcal{E}}{\partial t} & = \mathop{\rm curl}\mathcal{H} - \sigma \mathcal{E} \,, \\ \mu \frac{\partial \mathcal{H}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,, \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7190}% -$ \mathcal{E}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7192}% -$ \mathcal{H}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7194}% -$ \varepsilon (x), \mu(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7196}% -$ \sigma(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7202}% -$ L^{\infty}(\Omega )^{d\times d}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7204}% -$ \varepsilon (x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7206}% -$ \mu(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7211}% -$ \mathcal{E}(x,t)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7213}% -$ \mathcal{H}(x,t)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath7215}% -\begin{displaymath}\begin{split} \mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(i\omega t)\right) \,, \\ \mathcal{H}(x,t) &= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,. \end{split}\end{displaymath}% -\lthtmldisplayZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7217}% -$ E(x), H(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7219}% -$ \omega\neq 0$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7222}% -$ E(x) \exp(i\omega t)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7224}% -$ H(x) \exp(i\omega t)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7226}% -$ H(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7228}% -$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0 -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7230}% -$ |\omega|$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7232}% -$\displaystyle \omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,. -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7234}% -$ \omega^2\varepsilon E(x)$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7236}% -$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0 -$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_indisplay7242}% -$\displaystyle E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.$% -\lthtmlindisplaymathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7244}% -$ \tilde{E}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmlinlinemathA{tex2html_wrap_inline7246}% -$ \underline u = E - \tilde{E}$% -\lthtmlinlinemathZ -\lthtmlcheckvsize\clearpage} - -{\newpage\clearpage -\lthtmldisplayA{displaymath7248}% -\begin{displaymath}\begin{split} \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) + i\omega\sigma u &= F \quad \mathrm{in} \quad \Omega \,, \\ u \wedge n &= 0 \quad \mathrm{on} \quad \partial \Omega \,. 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a/deal.II/doc/reports/nedelec/index.html b/deal.II/doc/reports/nedelec/index.html deleted file mode 100644 index c7412e06ac..0000000000 --- a/deal.II/doc/reports/nedelec/index.html +++ /dev/null @@ -1,157 +0,0 @@ - - - - - -An -conforming FEM: Nédélec's elements of first type - - - - - - - - - - - - - - -

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

-

Anna Schneebeli, April 30, 2003

-

-Abstract: The aim of this report is to give an introduction to Nédélec's -$ H(\mathop {\rm curl};\Omega )$-conforming finite element method of first -type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [8]. -
-In the first section, we present the model problem and introduce the framework for its variational formulation. -
-In the second section, we present Nédélec's elements of first type for -$ H(\mathop {\rm curl};\Omega )$. -We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector -fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and -trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks -on approximation results. -
-Numerical results, which serve to illustrate the convergence of the method, are presented in the third section. -In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace -equation. -
-In Appendix B we motivate the model problem studied in the report by considering the -time-harmonic Maxwell's equations in the low-frequency case. - -

- -

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

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

-

Anna Schneebeli, April 30, 2003

-

-Abstract: The aim of this report is to give an introduction to Nédélec's -$ H(\mathop {\rm curl};\Omega )$-conforming finite element method of first -type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [8]. -
-In the first section, we present the model problem and introduce the framework for its variational formulation. -
-In the second section, we present Nédélec's elements of first type for -$ H(\mathop {\rm curl};\Omega )$. -We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector -fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and -trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks -on approximation results. -
-Numerical results, which serve to illustrate the convergence of the method, are presented in the third section. -In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace -equation. -
-In Appendix B we motivate the model problem studied in the report by considering the -time-harmonic Maxwell's equations in the low-frequency case. - -

- -

- - - - - -
-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node1.html b/deal.II/doc/reports/nedelec/node1.html deleted file mode 100644 index 90e43cab6e..0000000000 --- a/deal.II/doc/reports/nedelec/node1.html +++ /dev/null @@ -1,1018 +0,0 @@ - - - - - -1 The model problem and the space - - - - - - - - - - - - - - - -Subsections - - - -
- -

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

-

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

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

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

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

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

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

- -

-1.1 Definitions -

-

-

-

CONVENTION 1   -In the following, the vector -$ \underline t$ will denote the unit tangent vector w. r. t. an edge of a triangle or quadrilateral, - oriented counterclockwise with respect to the corresponding triangle or quadrilateral. - (In 3d, the considered triangles or quadrilaterals will always be faces of a polyhedron, and the counterclockwise orientation has to be - understood as induced by "outward unit normal of the face, plus right hand rule" ).

- -

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

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

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

-

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

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

-we have - -

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

-and - -

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

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

- -

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

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

-

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

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

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

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

-is a Hilbert space. -
 -

- -

-1.2 Trace theorem, integration by parts -

-The space -$ H(\mathop {\rm curl};\Omega )$ will be the appropriate Sobolev space for a weak formulation of the model problem. -In this section we provide a notion of trace of a -$ H(\mathop {\rm curl};\Omega )$-function onto the boundary -$ \partial \Omega $ and we define intergation by parts on the -space -$ H(\mathop {\rm curl};\Omega )$. - -

-

-

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

- -See [4] p.13, p.20 for the proof in the 2d-case and p.20 for a reference to the proof in 3d proposed in Duvaut & Lions, 1971. -
 -

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

-

-

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

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

- -

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

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

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

-PROOF. For smooth functions, it is easy to see that the above Green's formula holds. In the 2d case this follows just -from Gauss' divergence theorem and remark 1. -
-For the 3d case we use the identity - -

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

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

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

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

-

-

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

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

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

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

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

- -

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

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

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

-

DEFINITION 2   - -

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

- -

-

-

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

- -

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

-

-

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

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

- -

-1.3 Variational formulation of the model problem -

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

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

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

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

-are continuous and the bilinear form -$ a(\cdot,\cdot)$ is coercive on - -$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$. By the Lax-Milgram lemma it follows, that there exists a unique solution - -$ \underline u \in H_0(\mathop{\rm curl};\Omega )$ of (3). - -
-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node2.html b/deal.II/doc/reports/nedelec/node2.html deleted file mode 100644 index 1957a7a625..0000000000 --- a/deal.II/doc/reports/nedelec/node2.html +++ /dev/null @@ -1,4050 +0,0 @@ - - - - - -2 Nédélec's elements of first type for - - - - - - - - - - - - - - - -Subsections - - - -
- -

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

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

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

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

-First, we observe that for a conforming discretization of (3) we cannot take vector-valued finite elements - that are build by taking the standard nodal finite element spaces of globally continuous functions for each vector component. - For -$ H(\mathop {\rm curl};\Omega )$-functions, the only continuity condition is the continuity of the tangential component over cell boundaries. - This fact will motivate the choice of appropriate degrees of freedom (abbreviated by dofs in the following). -
-We will give an outline of the construction of the finite element spaces described by Nédélec in [8]. In literature, they are also referred to as - Nédélec's elements of first type. -
 -

- -

-2.1 Construction of Nédélec elements on tetrahedral grids -

-

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

- -

-2.1.1 Polynomial spaces on the reference element -

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

-

-

DEFINITION 3   -We define the auxiliary space -

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

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

- -

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

-

DEFINITION 4   -

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

-These spaces have dimension -

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

- -

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

-

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

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

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

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

-

-

EXAMPLE 1   - -

-

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

- -

- -

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

- -

-

- -

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

-

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

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

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

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

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

-
- - - - -
\begin{displaymath}\begin{split}&a_{11}=a_{22}=a_{33} = 0 \\ &a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,. \end{split}\end{displaymath} -   
-

- -

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

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

- -

-

- -

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

-

-

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

- -

- -

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

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

-

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

- -

-

-

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

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

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

-

-
inner dofs
-

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

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

-

-

- -

-

-

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

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

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

-

-
face dofs
-

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

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

-

-
inner dofs
-

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

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

- -

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

-

-

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

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

-

-

EXAMPLE 3 (Reference shape functions of lowest order for Nédélec elements on triangular meshes)   -Let the reference element be the triangle - -$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$. - Label the edges couterclockwise startung with -$ \hat{e}_0 = \overline{(0,0),(1,0)}$. - The tangential vectors to the edges are (oriented counterclockwise) - -

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

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

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

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

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

- -

- -

-
-2.1.3 Piola transformation -

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

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

-

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

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

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

-

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

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

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

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

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

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

-

- -

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

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

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

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

-

-

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

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

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

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

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

- -

-PROOF. -

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

-

-

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

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

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

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

- -

- -

- -

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

- In three dimensions, we cannot identify the curl-operator with the rotated - gradient or with the divergence of a rotated vector field. We cannot, as in 2d, derive a transformation formula for the curl from the - transformatin formula of the divergence. -
-By the chain rule, we obtain the transformation of the gradient of a vector field -$ \underline v$, defined by the Piola transformation ([*]) of a reference field -$ \hat{\underline v}$: -

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

-

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

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

-

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

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

-

-

-

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

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

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

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

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

- -

-PROOF. - It holds -

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

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

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

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

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

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

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

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

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

-

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

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

- -

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

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

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

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

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

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

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

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

- - -

- -

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

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

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

-

- -

-2.2.1 Polynomial spaces on the reference element -

-

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

-

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

- -

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

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

- -

-and in 3d -

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

- -

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

- -

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

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

-

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

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

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

-

-
inner dofs
-

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

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

-

-

- -

-

-

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

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

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

-

-
face dofs
-

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

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

-

-
inner dofs
-

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

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

- -

-

-

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

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

-they read -

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

- -

- -

-2.2.3 Transformation of the vector field -

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

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

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

- -

-2.3.1 Bilinear elements in 2d -

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

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

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

-

-

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

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

-as in the affine case. -

- -

-PROOF. - In this proof, the mapped element $ C$ will be fixed, so for simplicity we write $ F$ for $ F_C$. -
-First note that -$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$. We use the notation -$ D(F^{-1})_{ij}(x)= -\frac{\partial \hat{x}_i}{\partial x_j}(x)$ and imlicit summation to rewrite the Piola transformation of the vector field componentwise - -

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

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

-

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

-

-whereas for non-constant jacobian we have - -

-

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

-

-We see that in both cases we have - -

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

- that is, the second derivatives cancel out in the expression for the curl and the curl in the non-affine case transforms equally - to the curl in the affine case. - - -

- -

-2.3.2 Trilinear elements in 3d -

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

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

-

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

-

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

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

- -

- -

-2.4 Construction of global shape functions -

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

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

-

-

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

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

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

- -

-PROOF. - With - -

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

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

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

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

-As a consequence, we have - -

-

-

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

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

- -

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

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

- The following lemma will justify the choice of the moments describing the edge dofs. A consequence of the lemma will be, - that the matching of the local edge dofs -$ \alpha^{[K_+]}$ and -$ \alpha^{[K_-]}$ guarantees - the pointwise condition (26). -

-

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

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

- -

-PROOF. - -

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

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

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

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

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

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

-

-

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

- -

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

-

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

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

- -

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

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

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

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

- - -

-To close this section, let us make a note on the interpretation of the dofs on an element $ K$ in the case of lowest order polynomial - degree. In this case, all dofs are edge dofs, the degrees of freedom are -$ \hat{\alpha}_j(\hat{\underline -v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$ and the tangential traces of shape functions are constant on each edge. - Since we require -$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$ for the reference shape functions, we have - -

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

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

-

REMARK 6   -For the invariance of the edge dofs it is essential that the moments -$ \alpha^{[K]}$ on $ K$ are defined by using the - unit tangent - vector -$ \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert}\,(\hat{D}F)\,\hat{\underline t}$ on $ K$. If not, e. g. if we - just used the tangent -$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$, we would lose the invariance of the dofs. In that case the dofs would - scale by a factor depending on the size of the edge or face ([8], remark on p. 326). -

- -

- -

-2.5 Approximation and convergence results -

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

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

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

-

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

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

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

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

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

- Nédélec uses a standard scaling and Bramble-Hilbert argument to derive (29). Since - -$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$, the Bramble-Hilbert argument only guarantees an elementwise - approximation of order $ k$ of $ H^k(K)$-functions from the space -$ \mathcal{R}^k(K)$. -
-However, in a recent paper Hiptmair uses a duality technique to state optimal convergence of the - -$ L^2(\Omega )$-error -$ \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )}$ - for the 3d case and Nédélec's elements of first type of order $ k$ on tetrahedral meshes - (see Section 5.3, Theorem 5.8 in [5]): -

-

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

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

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

- - Several key arguments of the proof in [5] make explicitely use of features that are limited to 3d problems and the - family of finite elements based on tetrahedrons. - They cannot be modified trivially to apply to 2d problems or 3d problems on hexahedral meshes. Even worse, it is suggested by the results of - numerical experiments that one cannot hope to obtain a result similar to (30). -
-A possibility to overcome this deficiency of convergence is to use Nédélec elements of second type, where the full -$ [\mathbb{P}_k]^d$ - are used as polynomial spaces (see [10]). - -

- -

-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node3.html b/deal.II/doc/reports/nedelec/node3.html deleted file mode 100644 index 293cd62090..0000000000 --- a/deal.II/doc/reports/nedelec/node3.html +++ /dev/null @@ -1,723 +0,0 @@ - - - - - -3 Numerical results - - - - - - - - - - - - - - - -

-3 Numerical results -

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

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

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

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

- The finite element solution has been computed using Nédélec elements of first type and of polynomial degree $ k=1$ on a family of - affine triangular grids. - The initial coarse grid consisted of $ 2^5$ triangles. The finest grid with $ 2^{13}$ triangles results after five global refinements. -
-In Table 1 we see that for both examples we have - optimal convergence in the -$ H(\mathop {\rm curl};\Omega )$-semiorm, as we would expect from the theoretical results of the - previous section. As for the -$ L^2(\Omega )$-norm, it appears that in both examples the convergence - of the numerical solution is not optimal for our choice of finite elements. In the case of Nédélec elements of first - type and of polynomial degree $ k=1$, we got only -$ \mathcal{O}(h)$-convergence of the $ L^2$-error. However, this order of - convergence is consistent with the result (29) obtained by Nédélec in [8]. -
 -

-

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

-
-

- -

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

- -

- -

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

- -

-

-

REMARK 7   -The mesh generation and refinement was done by PDE-toolbox commands. Since the PDE-toolbox does not support three - dimensional grids, we restricted ourselves to 2d problems, and we have so far no numerical results for the case of - tetrahedral grids in 3d. -

- -

-As for meshes with quadrilateral cells, numerical results were obtained with a deal.II code, - using the finite element class fe/fenedelec.cc. This class provides - Nédélec's -$ H(\mathop {\rm curl};\Omega )$-conforming element of first type and lowest order in two and three space dimensions, on - bilinear quadrilateral, resp. trilinear hexahedral grids. For details about deal.II, see [2]. - In the following results were obtained for the model problem (1) in two dimensions using the - data (32). - We computed the solution on five successive - non-affine bilinear grids ( figure 3), each of which was obtained by global refinement of the previous one. - -

- -

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

- -

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

-

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

-
-

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

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

-In a first experiment, the finite element solution was computed on five successive globally refined affine grids. In a second -computation, we approximated the solution of the same problem on five successive globally refined non-affine trilinear grids. -
-We see in Table 3 that in both cases we observe again convergence of order -$ \mathcal{O}(h)$ in the - -$ H(\mathop {\rm curl};\Omega )$- and the -$ L^2(\Omega )$-norm. -
 -

-

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

-
-The conclusion that can be drawn from these numerical experiments is, that the restriction to three-dimensional tetrahedral grids -of Hiptmair's result on the $ L^2$-convergence of the error (6) cannot be relaxed. -
 -

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

- -

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

- -

- -

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

- - -

- -

-

-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node4.html b/deal.II/doc/reports/nedelec/node4.html deleted file mode 100644 index eab1c273dc..0000000000 --- a/deal.II/doc/reports/nedelec/node4.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - - -Bibliography - - - - - - - - - - - - - - - -

-Bibliography -

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

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

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

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

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

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

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

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

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

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

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

12 -
-W. Rachowicz and L. Demkowicz. -
A three-dimensional hp-adaptive finite element package for - electromagnetics (3Dhp90_EM). -
Ticam Report 00-04.2000, TICAM, 2000. -
Download version available on Demkowicz' webpage: - www.ices.utexas.edu/ Leszek. -
- -
-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node5.html b/deal.II/doc/reports/nedelec/node5.html deleted file mode 100644 index 7568b72569..0000000000 --- a/deal.II/doc/reports/nedelec/node5.html +++ /dev/null @@ -1,332 +0,0 @@ - - - - - -A. Construction of solutions in 2d - - - - - - - - - - - - - - - -

-A. Construction of solutions in 2d -

-

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

-

-

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

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

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

-
- - - - -
\begin{displaymath}\begin{split}\underline \mathop{\rm curl}\mathop{\rm curl}\un... -...e n = 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath} -   
-

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

- -

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

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

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

-

-

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

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

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

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

- -

-

-

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

- -

- - -

-
- -2003-04-30 -
- - diff --git a/deal.II/doc/reports/nedelec/node6.html b/deal.II/doc/reports/nedelec/node6.html deleted file mode 100644 index 138058a02f..0000000000 --- a/deal.II/doc/reports/nedelec/node6.html +++ /dev/null @@ -1,310 +0,0 @@ - - - - - -B. Time-harmonic Maxwell's equations with low-frequency approximation - - - - - - - - - - - - - - - -Subsections - - - -
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-B. Time-harmonic Maxwell's equations with low-frequency approximation -

-We show, how the model problem can be derived from the time-harmonic Maxwell's equations in the low-frequency case. -We follow the outline of [1]: -
-We consider the following primal formulation of Maxwell's equations: -

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

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

- -

-Time-harmonic, low-frequency case -

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

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

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

-

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

- -

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

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

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

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

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

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

-

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

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

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

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

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

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