+++ /dev/null
-\relax
-\bibstyle{abbrv}
+++ /dev/null
-\begin{thebibliography}{10}
-
-\bibitem{Alonso-Valli}
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- time-harmonic {M}axwell equations.
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-\bibitem{Deal}
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- Technical Reference}.
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-\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.
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-\bibitem{Girault-Raviart}
-V.~Girault and P.-A. Raviart.
-\newblock {\em Finite Element Approximation of the Navier-Stokes Equations},
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-\newblock Springer-Verlag, Berlin, Heidelberg, 1979, 1981.
-
-\bibitem{Hipt}
-R.~Hiptmair.
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- 2002.
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-\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
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-\bibitem{Ned1}
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-\newblock Mixed finite elements in $\mathbb{R}^3$.
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-
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-\newblock Elements finis mixtes incompressibles pour l'\'equation de {S}tokes
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-\newblock A new family of mixed finite elements in $\mathbb{R}^3$.
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-\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:
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-
-\bibitem{Demko3d}
-W.~Rachowicz and L.~Demkowicz.
-\newblock A three-dimensional hp-adaptive finite element package for
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-\newblock Ticam Report 00-04.2000, TICAM, 2000.
-\newblock Download version available on Demkowicz' webpage:
- www.ticam.utexas.edu/~{L}eszek.
-
-\end{thebibliography}
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-l.2552 ...ad \partial \Omega \,, ^^I^^I\end{split}
- \end{displaymath}%
-A left brace was mandatory here, so I've put one in.
-You might want to delete and/or insert some corrections
-so that I will find a matching right brace soon.
-(If you're confused by all this, try typing `I}' now.)
-
-! Missing } inserted.
-<inserted text>
- }
-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.
-
-! Missing { inserted.
-<to be read again>
- }
-l.2552 ...ad \partial \Omega \,, ^^I^^I\end{split}
- \end{displaymath}%
-A left brace was mandatory here, so I've put one in.
-You might want to delete and/or insert some corrections
-so that I will find a matching right brace soon.
-(If you're confused by all this, try typing `I}' now.)
-
-! Missing } inserted.
-<inserted text>
- }
-l.2552 ...ad \partial \Omega \,, ^^I^^I\end{split}
- \end{displaymath}%
-I've put in what seems to be necessary to fix
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-<to be read again>
- }
-l.2590 $ \underline \mathop
- {\rm curl}\mathop{\rm curl}\underline E = \nabla(...
-A left brace was mandatory here, so I've put one in.
-You might want to delete and/or insert some corrections
-so that I will find a matching right brace soon.
-(If you're confused by all this, try typing `I}' now.)
-
-! Missing } inserted.
-<inserted text>
- }
-l.2590 ...\cdot\underline E) - \Delta\underline E$
- %
-I've inserted something that you may have forgotten.
-(See the <inserted text> above.)
-With luck, this will get me unwedged. But if you
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-
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-! Missing { inserted.
-<to be read again>
- }
-l.2596 $ \underline \mathop
- {\rm curl}\mathop{\rm curl}\underline E = - \Delt...
-A left brace was mandatory here, so I've put one in.
-You might want to delete and/or insert some corrections
-so that I will find a matching right brace soon.
-(If you're confused by all this, try typing `I}' now.)
-
-! Missing } inserted.
-<inserted text>
- }
-l.2596 ...url}\underline E = - \Delta\underline E$
- %
-I've inserted something that you may have forgotten.
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-$\displaystyle H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d: \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
-$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay5525}%
-$\displaystyle (\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\underline v, \underline u)_{L^2(\Omega )} + (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
-$%
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-\lthtmlcheckvsize\clearpage}
-
-\stepcounter{subsection}
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5539}%
-$ [\mathcal{D}(\overline{\Omega })]^d$%
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-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5546}%
-$ \underline u$%
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-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5548}%
-$ [H(\mathop{\rm curl};\Omega )]^2$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5552}%
-$ H^1(\Omega )$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay5554}%
-$\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$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5562}%
-$ [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$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay5572}%
-$\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$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay5591}%
-$\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$%
-\lthtmlinlinemathZ
-\lthtmlcheckvsize\clearpage}
-
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-$\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$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline5946}%
-$ \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$%
-\lthtmlinlinemathZ
-\lthtmlcheckvsize\clearpage}
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-$ m$%
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-$ \mathcal{Q}_{l,m,n}$%
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-$\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\}\,,$%
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-$\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\}\,.$%
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-
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-
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-$\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})\,,$%
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-
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-$\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}\,.$%
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-
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-\lthtmlinlinemathA{tex2html_wrap_inline6273}%
-$ 12k$%
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-
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-$\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})\,.$%
-\lthtmlindisplaymathZ
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-
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-$ 6\cdot 2k(k-1)$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay6283}%
-$\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}\,.$%
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-
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-$ 3k(k-1)^2$%
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-\lthtmlcheckvsize\clearpage}
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-$ [0,1]^2$%
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-
-{\newpage\clearpage
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-$\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)\,,
-$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
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-$\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)\,.$%
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-
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-$ F_C(\hat{C})$%
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-
-{\newpage\clearpage
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-$ \hat{D}F_C(\hat{x})$%
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-
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-$ F_C$%
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-$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$%
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-
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-$\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})\,,$%
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-
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-$ F$%
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-
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-$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$%
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-
-{\newpage\clearpage
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-$ D(F^{-1})_{ij}(x)=
-\frac{\partial \hat{x}_i}{\partial x_j}(x)$%
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-
-{\newpage\clearpage
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-$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
-$%
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-
-{\newpage\clearpage
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-\begin{displaymath}\begin{split} \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\ \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath}%
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-\begin{displaymath}\begin{split} \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) + \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\ \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) + \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath}%
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-$\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)) \,
-$%
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-
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-
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-
-{\newpage\clearpage
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-$\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)
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-$%
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-
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-
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-$ \hat{D}F_C(\hat(x))$%
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-
-{\newpage\clearpage
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-
-{\newpage\clearpage
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-$\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}\,.$%
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-
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
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-$ \mathcal{P}_K$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
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-$ K=F(\hat{K})$%
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-
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-$ [0,|e|] \ni s \mapsto \underline x(s) \in e$%
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-
-{\newpage\clearpage
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-$ [0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
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-$ \frac{d \underline x}{ds}$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6413}%
-$ \frac{d \hat{\underline x}}{d\hat{s}}$%
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-
-{\newpage\clearpage
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-$\displaystyle \underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,$%
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-
-{\newpage\clearpage
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-$ |\hat{e}|$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
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-$ |e|$%
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-
-{\newpage\clearpage
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-$\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})\,
-$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6434}%
-$ \hat{x}_j = \hat{x}_j(\underline x(s))$%
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-
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-$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$%
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-
-{\newpage\clearpage
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-$\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}
-$%
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-
-{\newpage\clearpage
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-$ \frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$%
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-
-{\newpage\clearpage
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-$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$%
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-
-{\newpage\clearpage
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-$\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)\,.
-$%
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-
-{\newpage\clearpage
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-$ K_- = F_-(\hat{K})$%
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-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6457}%
-$ K_+ = F_+(\hat{K})$%
-\lthtmlinlinemathZ
-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6461}%
-$ \underline N$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6465}%
-$ \underline N_-$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6467}%
-$ \underline N_+$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6475}%
-$ e_+ =F_+(\hat{e}_i)$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6477}%
-$ e_- =F_-(\hat{e}_j)$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6479}%
-$ \underline t_+ $%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6485}%
-$ \underline t_- = -\underline t_+$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6491}%
-$ \int_{e_+}$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6495}%
-$ \int_{e_-}$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_inline6499}%
-$ \underline t_-$%
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-\lthtmlcheckvsize\clearpage}
-
-{\newpage\clearpage
-\lthtmlinlinemathA{tex2html_wrap_indisplay6503}%
-$\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)$%
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-\lthtmlcheckvsize\clearpage}
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-
-<!--Converted with LaTeX2HTML 2K.1beta (1.47)
-original version by: Nikos Drakos, CBLU, University of Leeds
-* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
-* with significant contributions from:
- Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
-<HTML>
-<HEAD>
-<TITLE>An -conforming FEM: Nédélec's elements of first type</TITLE>
-<META NAME="description" CONTENT="An -conforming FEM: Nédélec's elements of first type">
-<META NAME="keywords" CONTENT="main">
-<META NAME="resource-type" CONTENT="document">
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-</HEAD>
-
-<BODY >
-<H1 ALIGN=CENTER><FONT SIZE="+2">An <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM:
-<BR>
-Nédélec's elements of first type</FONT></H1>
-<P ALIGN=CENTER><STRONG><FONT SIZE="-1">Anna Schneebeli, April 30, 2003</FONT></STRONG></P>
-<P ALIGN=LEFT></P>
-<FONT SIZE="-1"><B>Abstract:</B> The aim of this report is to give an introduction to Nédélec's <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method of first
-type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [<A
- HREF="node4.html#Ned1">8</A>].
-<BR>
-In the first section, we present the model problem and introduce the framework for its variational formulation.
-<BR>
-In the second section, we present Nédélec's elements of first type for <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">.
-We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector
-fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and
-trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks
-on approximation results.
-<BR>
-Numerical results, which serve to illustrate the convergence of the method, are presented in the third section.
-In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace
-equation.
-<BR>
-In Appendix B we motivate the model problem studied in the report by considering the
-time-harmonic Maxwell's equations in the low-frequency case.
-</FONT>
-<P>
-<FONT SIZE="-1"></FONT>
-<BR><HR>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"></A>
-
-<UL>
-<LI><A NAME="tex2html11"
- HREF="node1.html">1 <FONT SIZE="+1">The model problem and the space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-<UL>
-<LI><A NAME="tex2html12"
- HREF="node1.html#SECTION00011000000000000000">1.1 Definitions</A>
-<LI><A NAME="tex2html13"
- HREF="node1.html#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
-<LI><A NAME="tex2html14"
- HREF="node1.html#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html15"
- HREF="node2.html">2 <FONT SIZE="+1">Nédélec's elements of first type for <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-<UL>
-<LI><A NAME="tex2html16"
- HREF="node2.html#SECTION00021000000000000000">2.1 Construction of Nédélec elements on tetrahedral grids</A>
-<UL>
-<LI><A NAME="tex2html17"
- HREF="node2.html#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html18"
- HREF="node2.html#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html19"
- HREF="node2.html#SECTION00021300000000000000">2.1.3 Piola transformation</A>
-<LI><A NAME="tex2html20"
- HREF="node2.html#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
-<LI><A NAME="tex2html21"
- HREF="node2.html#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
-</UL>
-<LI><A NAME="tex2html22"
- HREF="node2.html#SECTION00022000000000000000">2.2 Nédélec Elements on affine quadrilateral or hexahedral grids</A>
-<UL>
-<LI><A NAME="tex2html23"
- HREF="node2.html#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html24"
- HREF="node2.html#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html25"
- HREF="node2.html#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
-</UL>
-<LI><A NAME="tex2html26"
- HREF="node2.html#SECTION00023000000000000000">2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
-<UL>
-<LI><A NAME="tex2html27"
- HREF="node2.html#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
-<LI><A NAME="tex2html28"
- HREF="node2.html#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
-</UL>
-<LI><A NAME="tex2html29"
- HREF="node2.html#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
-<LI><A NAME="tex2html30"
- HREF="node2.html#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html31"
- HREF="node3.html">3 <FONT SIZE="+1">Numerical results</FONT></A>
-<LI><A NAME="tex2html32"
- HREF="node4.html">Bibliography</A>
-<LI><A NAME="tex2html33"
- HREF="node5.html">A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
-<LI><A NAME="tex2html34"
- HREF="node6.html">A. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT></A>
-<UL>
-<LI><A NAME="tex2html35"
- HREF="node6.html#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
-</UL></UL>
-<!--End of Table of Child-Links-->
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
+++ /dev/null
-/* Century Schoolbook font is very similar to Computer Modern Math: cmmi */
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+++ /dev/null
-<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
-
-<!--Converted with LaTeX2HTML 2K.1beta (1.47)
-original version by: Nikos Drakos, CBLU, University of Leeds
-* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
-* with significant contributions from:
- Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
-<HTML>
-<HEAD>
-<TITLE>An -conforming FEM: Nédélec's elements of first type</TITLE>
-<META NAME="description" CONTENT="An -conforming FEM: Nédélec's elements of first type">
-<META NAME="keywords" CONTENT="main">
-<META NAME="resource-type" CONTENT="document">
-<META NAME="distribution" CONTENT="global">
-
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-<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
-
-<LINK REL="STYLESHEET" HREF="main.css">
-
-</HEAD>
-
-<BODY >
-<H1 ALIGN=CENTER><FONT SIZE="+2">An <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM:
-<BR>
-Nédélec's elements of first type</FONT></H1>
-<P ALIGN=CENTER><STRONG><FONT SIZE="-1">Anna Schneebeli, April 30, 2003</FONT></STRONG></P>
-<P ALIGN=LEFT></P>
-<FONT SIZE="-1"><B>Abstract:</B> The aim of this report is to give an introduction to Nédélec's <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method of first
-type. As the name suggests, this method has been introduced in 1980 by J. C. Nédélec in [<A
- HREF="node4.html#Ned1">8</A>].
-<BR>
-In the first section, we present the model problem and introduce the framework for its variational formulation.
-<BR>
-In the second section, we present Nédélec's elements of first type for <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">.
-We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector
-fields and discuss the choice of function spaces and degrees of freedom. These results are then extendend to bi- and
-trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks
-on approximation results.
-<BR>
-Numerical results, which serve to illustrate the convergence of the method, are presented in the third section.
-In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace
-equation.
-<BR>
-In Appendix B we motivate the model problem studied in the report by considering the
-time-harmonic Maxwell's equations in the low-frequency case.
-</FONT>
-<P>
-<FONT SIZE="-1"></FONT>
-<BR><HR>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"></A>
-
-<UL>
-<LI><A NAME="tex2html11"
- HREF="node1.html">1 <FONT SIZE="+1">The model problem and the space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-<UL>
-<LI><A NAME="tex2html12"
- HREF="node1.html#SECTION00011000000000000000">1.1 Definitions</A>
-<LI><A NAME="tex2html13"
- HREF="node1.html#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
-<LI><A NAME="tex2html14"
- HREF="node1.html#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html15"
- HREF="node2.html">2 <FONT SIZE="+1">Nédélec's elements of first type for <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-<UL>
-<LI><A NAME="tex2html16"
- HREF="node2.html#SECTION00021000000000000000">2.1 Construction of Nédélec elements on tetrahedral grids</A>
-<UL>
-<LI><A NAME="tex2html17"
- HREF="node2.html#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html18"
- HREF="node2.html#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html19"
- HREF="node2.html#SECTION00021300000000000000">2.1.3 Piola transformation</A>
-<LI><A NAME="tex2html20"
- HREF="node2.html#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
-<LI><A NAME="tex2html21"
- HREF="node2.html#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
-</UL>
-<LI><A NAME="tex2html22"
- HREF="node2.html#SECTION00022000000000000000">2.2 Nédélec Elements on affine quadrilateral or hexahedral grids</A>
-<UL>
-<LI><A NAME="tex2html23"
- HREF="node2.html#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html24"
- HREF="node2.html#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html25"
- HREF="node2.html#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
-</UL>
-<LI><A NAME="tex2html26"
- HREF="node2.html#SECTION00023000000000000000">2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
-<UL>
-<LI><A NAME="tex2html27"
- HREF="node2.html#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
-<LI><A NAME="tex2html28"
- HREF="node2.html#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
-</UL>
-<LI><A NAME="tex2html29"
- HREF="node2.html#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
-<LI><A NAME="tex2html30"
- HREF="node2.html#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html31"
- HREF="node3.html">3 <FONT SIZE="+1">Numerical results</FONT></A>
-<LI><A NAME="tex2html32"
- HREF="node4.html">Bibliography</A>
-<LI><A NAME="tex2html33"
- HREF="node5.html">A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
-<LI><A NAME="tex2html34"
- HREF="node6.html">A. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT></A>
-<UL>
-<LI><A NAME="tex2html35"
- HREF="node6.html#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
-</UL></UL>
-<!--End of Table of Child-Links-->
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
+++ /dev/null
-<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
-
-<!--Converted with LaTeX2HTML 2K.1beta (1.47)
-original version by: Nikos Drakos, CBLU, University of Leeds
-* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
-* with significant contributions from:
- Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
-<HTML>
-<HEAD>
-<TITLE>1 The model problem and the space </TITLE>
-<META NAME="description" CONTENT="1 The model problem and the space ">
-<META NAME="keywords" CONTENT="main">
-<META NAME="resource-type" CONTENT="document">
-<META NAME="distribution" CONTENT="global">
-
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-<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
-
-<LINK REL="STYLESHEET" HREF="main.css">
-
-</HEAD>
-
-<BODY >
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
-
-<UL>
-<LI><A NAME="tex2html46"
- HREF="#SECTION00011000000000000000">1.1 Definitions</A>
-<LI><A NAME="tex2html47"
- HREF="#SECTION00012000000000000000">1.2 Trace theorem, integration by parts</A>
-<LI><A NAME="tex2html48"
- HREF="#SECTION00013000000000000000">1.3 Variational formulation of the model problem</A>
-</UL>
-<!--End of Table of Child-Links-->
-<HR>
-
-<H1><A NAME="SECTION00010000000000000000">
-1 <FONT SIZE="+1">The model problem and the space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-</H1>
-<P>
-<FONT SIZE="-1">Consider the vector-valued model problem in a Lipschitz domain <!-- MATH
- $\Omega \in \mathbb{R}^d$
- -->
-<IMG
- WIDTH="61" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img4.gif"
- ALT="$ \Omega \in \mathbb{R}^d$">, <IMG
- WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$ d=2,3$">:
-</FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:model_problem"></A><!-- MATH
- \begin{equation}
-\mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u = \underline f \quad \mathrm{in} \quad \Omega \,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="246" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img6.gif"
- ALT="$\displaystyle \mathop{\rm curl}\mathop{\rm curl}\underline u + c(x) \underline u = \underline f \quad \mathrm{in} \quad \Omega \,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(1)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-with right hand side <!-- MATH
- $\underline f \in L^2(\Omega )^d$
- -->
-<IMG
- WIDTH="92" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img7.gif"
- ALT="$ \underline f \in L^2(\Omega )^d$">.
-<BR>
-We assume a homogeneous Dirichlet boundary condition on the tangential trace
-</FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_PCB"></A><!-- MATH
- \begin{equation}
-\underline u \wedge \underline n = 0
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img8.gif"
- ALT="$\displaystyle \underline u \wedge \underline n = 0$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(2)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-on the boundary <!-- MATH
- $\partial \Omega$
- -->
-<IMG
- WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img9.gif"
- ALT="$ \partial \Omega $"> of <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $">.
-<BR>
-The coefficient <IMG
- WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img11.gif"
- ALT="$ c(x)$"> is assumed to be bounded and uniform positive definite.
-<BR>
-This type of problem typically arises in particular settings of
-Maxwell`s equations. The boundary condition (<A HREF="node1.html#eq:_PCB">2</A>) then applies to a perfectly conducting boundary.
-For a derivation of the model problem (<A HREF="node1.html#eq:model_problem">1</A>), refer to Appendix <A HREF="node6.html#appendix:_Maxwell_s_Eq.">B</A>.
-<BR></FONT>
-<P>
-<FONT SIZE="-1">The subject of this section is to give an appropriate setting for a variational formulation of (<A HREF="node1.html#eq:model_problem">1</A>).
-<BR>
-A more detailed treatment of the following notions and proofs can be found in [<A
- HREF="node4.html#Girault-Raviart">4</A>].
-</FONT>
-<P>
-
-<H2><A NAME="SECTION00011000000000000000">
-1.1 Definitions</A>
-</H2>
-<P>
-<P>
-<DIV><A NAME="def:_tangent"><B>C<SMALL>ONVENTION</SMALL> 1</B></A>
-In the following, the vector <!-- MATH
- $\underline t$
- -->
-<IMG
- WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img12.gif"
- ALT="$ \underline t$"> will denote the unit tangent vector w. r. t. an edge of a triangle or quadrilateral,
- oriented counterclockwise with respect to the corresponding triangle or quadrilateral.
- (In 3d, the considered triangles or quadrilaterals will always be faces of a polyhedron, and the counterclockwise orientation has to be
- understood as induced by "outward unit normal of the face, plus right hand rule" ).</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">Let us first consider the case of <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$">. For <!-- MATH
- $\underline v = \left(\begin{array}{c} v_1(x,y) \\v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$
- -->
-<IMG
- WIDTH="215" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img14.gif"
- ALT="$ \underline v = \left(\begin{array}{c} v_1(x,y) \\ v_2(x,y)\end{array} \right) \in [\mathcal{D}(\overline{\Omega })]^2$">
-and <!-- MATH
- $\varphi \in \mathcal{D}(\overline{\Omega })$
- -->
-<IMG
- WIDTH="79" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img15.gif"
- ALT="$ \varphi \in \mathcal{D}(\overline{\Omega })$"> we define the scalar- and the
-vector-valued curl-operators:
-</FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_1 \quad \mathrm{and} \quad \mathop{\underline{\rm curl}}\varphi := \left(\begin{array}{c} \partial _y\varphi \\-\partial _x\varphi \end{array}
- \right) \,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="396" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img16.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v := \partial _x v_2 - \partial _y v_...
-...in{array}{c} \partial _y\varphi \\ -\partial _x\varphi \end{array}\right) \,.
-$">
-</DIV><P></P><FONT SIZE="-1">
-We note that the <!-- MATH
- $\mathop{\rm curl}\mathop{\rm curl}$
- -->
-<IMG
- WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img17.gif"
- ALT="$ \mathop{\rm curl}\mathop{\rm curl}$">-operator in two dimensions has to be understood as <!-- MATH
- $\mathop{\underline{\rm curl}}\mathop{\rm curl}$
- -->
-<IMG
- WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img18.gif"
- ALT="$ \mathop{\underline{\rm curl}}\mathop{\rm curl}$">.
-<BR></FONT><P>
-<DIV><A NAME="re:_rotation"><B>R<SMALL>EMARK</SMALL> 1</B></A>
-In the two dimensional case, the <!-- MATH
- $\mathop{\rm curl}$
- -->
-<IMG
- WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img19.gif"
- ALT="$ \mathop{\rm curl}$"> operator is simply the divergence of the rotated field <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$">. Similarly, the <!-- MATH
- $\mathop{\underline{\rm curl}}$
- -->
-<IMG
- WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img21.gif"
- ALT="$ \mathop{\underline{\rm curl}}$"> operator is
-the rotated gradient field of <IMG
- WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img22.gif"
- ALT="$ \varphi $">. Setting
-<!-- MATH
- \begin{displaymath}
-\boldsymbol{R} = \left(\begin{array}{cc}
- 0 & 1 \\
- -1 & 0
- \end{array}\right) \,,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="144" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img23.gif"
- ALT="$\displaystyle \boldsymbol{R} = \left(\begin{array}{cc}
-0 & 1 \\
--1 & 0
-\end{array}\right) \,,
-$">
-</DIV><P></P>
-we have
-<!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right)
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img24.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \mathrm{div} \left(\boldsymbol{R} \underline v\right)
-$">
-</DIV><P></P>
-and
-<!-- MATH
- \begin{displaymath}
-\mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="122" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img25.gif"
- ALT="$\displaystyle \mathop{\underline{\rm curl}}\varphi = \boldsymbol{R} \nabla\varphi \,.
-$">
-</DIV><P></P>
-We further note that the tangential vector <!-- MATH
- $\underline t$
- -->
-<IMG
- WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img12.gif"
- ALT="$ \underline t$"> is just the rotated outward unit normal vector
-<!-- MATH
- $\underline t = \boldsymbol{R}^T\underline n$
- -->
-<IMG
- WIDTH="72" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img26.gif"
- ALT="$ \underline t = \boldsymbol{R}^T\underline n$">. This will enable us to derive statements for the <!-- MATH
- $\mathop{\rm curl}$
- -->
-<IMG
- WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img19.gif"
- ALT="$ \mathop{\rm curl}$">-operators in two dimensions from statements for the
-divergence and gradient operators in two dimensions.</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">For the case of <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$"> and a vector field <!-- MATH
- $\underline v \in [\mathcal{D}(\overline{\Omega })]^3$
- -->
-<IMG
- WIDTH="94" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img28.gif"
- ALT="$ \underline v \in [\mathcal{D}(\overline{\Omega })]^3$"> we write
-</FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}\underline v := \nabla \wedge \underline v := \left(\begin{array}{c}
- \partial _y v_3 - \partial _z v_2 \\
- \partial _z v_1 - \partial _x v_3 \\
- \partial _x v_2 - \partial _y v_1
- \end{array} \right)
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="281" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
- SRC="img29.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v := \nabla \wedge \underline v := \l...
-... - \partial _x v_3 \\
-\partial _x v_2 - \partial _y v_1
-\end{array} \right)
-$">
-</DIV><P></P><P>
-<DIV><B>D<SMALL>EFINITION</SMALL> 1</B>
-For <IMG
- WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$ d=2,3$"> we write <!-- MATH
- $\tilde{d}=1$
- -->
-<IMG
- WIDTH="47" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img30.gif"
- ALT="$ \tilde{d}=1$"> if <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$"> and <!-- MATH
- $\tilde{d}=3$
- -->
-<IMG
- WIDTH="47" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img31.gif"
- ALT="$ \tilde{d}=3$"> if <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$">, and we define
-<!-- MATH
- \begin{displaymath}
-H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d: \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="364" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
- SRC="img32.gif"
- ALT="$\displaystyle H(\mathop{\rm curl}; \Omega ) := \{ \underline v \in [L^2(\Omega )]^d: \mathop{\rm curl}\underline v \in [L^2(\Omega )]^{\tilde{d}} \}
-$">
-</DIV><P></P></DIV><P></P>
-<FONT SIZE="-1">
-<!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"> endowed with the inner product
-</FONT><!-- MATH
- \begin{displaymath}
-(\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\underline v, \underline u)_{L^2(\Omega )} + (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="374" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img33.gif"
- ALT="$\displaystyle (\underline v, \underline u)_{H(\mathop{\rm curl};\Omega )} := (\...
-... (\mathop{\rm curl}\underline v, \mathop{\rm curl}\underline u)_{L^2(\Omega )}
-$">
-</DIV><P></P><FONT SIZE="-1">
-is a Hilbert space.
-<BR></FONT>
-<P>
-
-<H2><A NAME="SECTION00012000000000000000">
-1.2 Trace theorem, integration by parts</A>
-</H2><FONT SIZE="-1">
-The space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"> will be the appropriate Sobolev space for a weak formulation of the model problem.
-In this section we provide a notion of trace of a <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-function onto the boundary <!-- MATH
- $\partial \Omega$
- -->
-<IMG
- WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img9.gif"
- ALT="$ \partial \Omega $"> and we define intergation by parts on the
-space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">.
-</FONT>
-<P>
-<P>
-<DIV><A NAME="th:density"><B>T<SMALL>HEOREM</SMALL> 1</B></A> (Approximation Property)
-For <IMG
- WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$ d=2,3$">, <!-- MATH
- $[\mathcal{D}(\overline{\Omega })]^d$
- -->
-<IMG
- WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img34.gif"
- ALT="$ [\mathcal{D}(\overline{\Omega })]^d$"> is dense in <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">.</DIV><P></P>
-<FONT SIZE="-1">
-See [<A
- HREF="node4.html#Girault-Raviart">4</A>] p.13, p.20 for the proof in the 2d-case and p.20 for a reference to the proof in 3d proposed in Duvaut & Lions, 1971.
-<BR></FONT>
-<P>
-<FONT SIZE="-1">Equipped with this approximation property of smooth functions to elements of <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">, we can state
-</FONT>
-<P>
-<P>
-<DIV><A NAME="eq:_PI"><B>T<SMALL>HEOREM</SMALL> 2</B></A> (Green's Formula)
-For the 2d case, let <!-- MATH
- $\underline u$
- -->
-<IMG
- WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img35.gif"
- ALT="$ \underline u$"> be in <!-- MATH
- $[H(\mathop{\rm curl};\Omega )]^2$
- -->
-<IMG
- WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img36.gif"
- ALT="$ [H(\mathop{\rm curl};\Omega )]^2$"> and <IMG
- WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img22.gif"
- ALT="$ \varphi $"> be a test function in <!-- MATH
- $H^1(\Omega )$
- -->
-<IMG
- WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img37.gif"
- ALT="$ H^1(\Omega )$">. We have
-<!-- MATH
- \begin{displaymath}
-\int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx = \int_{\Omega } \underline u \cdot \mathop{\underline{\rm curl}}\varphi \,dx + \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="380" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img38.gif"
- ALT="$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u \,\,\varphi \,dx = \...
-...+ \int_{\partial \Omega } (\underline u\cdot \underline t) \, \varphi \,ds \:,
-$">
-</DIV><P></P>
-
-<P>
-For the 3d case, let <!-- MATH
- $\underline u$
- -->
-<IMG
- WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img35.gif"
- ALT="$ \underline u$"> be in <!-- MATH
- $[H(\mathop{\rm curl};\Omega )]^3$
- -->
-<IMG
- WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img39.gif"
- ALT="$ [H(\mathop{\rm curl};\Omega )]^3$"> and <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$"> be a test function in <!-- MATH
- $[H^1(\Omega )]^3$
- -->
-<IMG
- WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img40.gif"
- ALT="$ [H^1(\Omega )]^3$">. We then have
-<!-- MATH
- \begin{displaymath}
-\int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \,dx = \int_{\Omega } \underline u \cdot \mathop{\rm curl}\underline v \,dx + \int_{\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="400" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img41.gif"
- ALT="$\displaystyle \int_{\Omega } \underline v \cdot \mathop{\rm curl}\underline u \...
-...\partial \Omega } (\underline v\wedge \underline n) \cdot \underline u\,ds \:,
-$">
-</DIV><P></P>
-The boundary integrals are understood as duality pairings in <!-- MATH
- $[H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$
- -->
-<IMG
- WIDTH="181" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img42.gif"
- ALT="$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}} \times H^{\frac{1}{2}}(\partial \Omega )$"> .
-<BR></DIV><P></P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>For smooth functions, it is easy to see that the above Green's formula holds. In the 2d case this follows just
-from Gauss' divergence theorem and remark <A HREF="node1.html#re:_rotation">1</A>.
-<BR>
-For the 3d case we use the identity
-</FONT><!-- MATH
- \begin{displaymath}
-\mathrm{div}\, (\underline u \wedge \underline v) = \underline v \cdot \mathop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="257" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img43.gif"
- ALT="$\displaystyle \mathrm{div}\, (\underline u \wedge \underline v) = \underline v ...
-...thop{\rm curl}\underline u - \underline u \cdot \mathop{\rm curl}\underline v
-$">
-</DIV><P></P><FONT SIZE="-1">
-together with Gauss' Divergence Theorem and the properties of the mixed product <!-- MATH
- $(\underline a\wedge\underline b)\cdot \underline c$
- -->
-<IMG
- WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img44.gif"
- ALT="$ (\underline a\wedge\underline b)\cdot \underline c$"> to obtain
-</FONT><!-- MATH
- \begin{displaymath}
-\int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - \underline u\cdot \mathop{\rm curl}\underline v \, dx = \int_{\Omega } \mathrm{div}\, (\underline u \wedge \underline v) \, dx
- = \int_{\partial \Omega } (\underline u \wedge \underline v)\cdot \underline n \,ds = \int_{\partial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="649" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
- SRC="img45.gif"
- ALT="$\displaystyle \int_{\Omega } \underline v\cdot \mathop{\rm curl}\underline u - ...
-...rtial \Omega } (\underline v \wedge \underline n)\cdot \underline u \,ds \,\,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- The extention to a pairing of
- <!-- MATH
- $H(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img46.gif"
- ALT="$ H(\mathop{\rm curl})$"> and <!-- MATH
- $H^1(\Omega )$
- -->
-<IMG
- WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img37.gif"
- ALT="$ H^1(\Omega )$"> functions follows with Theorem <A HREF="node1.html#th:density">1</A> by a density argument and is a result of the proof of the
- Trace Theorem. See [<A
- HREF="node4.html#Girault-Raviart">4</A>] p.21 for details.
-
-</FONT>
-<P>
-<P>
-<DIV><B>T<SMALL>HEOREM</SMALL> 3</B> (Trace Theorem)
-For <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$">, let <!-- MATH
- $\underline n$
- -->
-<IMG
- WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img47.gif"
- ALT="$ \underline n$"> denote the outward unit normal to the boundary <!-- MATH
- $\partial \Omega$
- -->
-<IMG
- WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img9.gif"
- ALT="$ \partial \Omega $">. For <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$">, let <!-- MATH
- $\underline t$
- -->
-<IMG
- WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img12.gif"
- ALT="$ \underline t$"> be as in convention <A HREF="node1.html#def:_tangent">1</A>
-<BR>
-For <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$"> the mapping
-<!-- MATH
- \begin{displaymath}
-\gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img48.gif"
- ALT="$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \cdot \underline t
-$">
-</DIV><P></P>
-and for <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$"> the mapping
-<!-- MATH
- \begin{displaymath}
-\gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img49.gif"
- ALT="$\displaystyle \gamma: \quad \underline v \mapsto \gamma(\underline v) \wedge \underline n
-$">
-</DIV><P></P>
-is contiuous and linear from <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"> to <!-- MATH
- $[H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$
- -->
-<IMG
- WIDTH="97" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img50.gif"
- ALT="$ [H^{-\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$">.</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">Note, that the trace of a <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-function is only defined in tangential direction. Its trace is in the dual space
- of traces of <!-- MATH
- $[H^1(\Omega )]^{\tilde{d}}$
- -->
-<IMG
- WIDTH="74" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img51.gif"
- ALT="$ [H^1(\Omega )]^{\tilde{d}}$"> functions. Recall that traces of such functions
-are defined in every direction and are functions in <!-- MATH
- $[H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$
- -->
-<IMG
- WIDTH="87" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img52.gif"
- ALT="$ [H^{\frac{1}{2}}(\partial \Omega )]^{\tilde{d}}$">.
-<BR></FONT>
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
-The proof of the trace theorem follows from Green's formula stated in theorem <A HREF="node1.html#eq:_PI">2</A> applied to smooth functions
-and then by density arguments. See [<A
- HREF="node4.html#Girault-Raviart">4</A>] p.21 for details.
-
-</FONT>
-<P>
-<FONT SIZE="-1">Due to the Trace Theorem it makes sense to define a space of <!-- MATH
- $H(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img46.gif"
- ALT="$ H(\mathop{\rm curl})$">-functions with vanishing tangential components on the boundary.
- </FONT><P>
-<DIV><B>D<SMALL>EFINITION</SMALL> 2</B>
-<!-- MATH
- \begin{displaymath}
-H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mathop{\rm curl};\Omega ): \quad \underline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\}
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="406" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img53.gif"
- ALT="$\displaystyle H_0(\mathop{\rm curl};\Omega ) := \left\{ \underline v \in H(\mat...
-...nderline v\wedge \underline n = 0 \:\:\mathrm{on}\:\: \partial \Omega \right\}
-$">
-</DIV><P></P></DIV><P></P>
-
-<P>
-<P>
-<DIV><B>R<SMALL>EMARK</SMALL> 2</B>
-For <IMG
- WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$ d=2,3$">, <!-- MATH
- $[\mathcal{D}(\Omega )]^d$
- -->
-<IMG
- WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img54.gif"
- ALT="$ [\mathcal{D}(\Omega )]^d$"> is dense in <!-- MATH
- $H_0(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img55.gif"
- ALT="$ H_0(\mathop{\rm curl};\Omega )$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">A consequence of Green's formula is the following important regularity property of <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-functions:
-</FONT>
-<P>
-<P>
-<DIV><A NAME="prop:_no_jumps"><B>P<SMALL>ROPOSITION</SMALL> 1</B></A>
-Let <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$"> and <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> be two polygonal (resp. polyhedral) Lipschitz domains in <!-- MATH
- $\mathbb{R}^d$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img58.gif"
- ALT="$ \mathbb{R}^d$">, with a common edge (resp. common edge or face)
- <!-- MATH
- $e = \partial K_-\cap\partial K_+ \neq \emptyset$
- -->
-<IMG
- WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img59.gif"
- ALT="$ e = \partial K_-\cap\partial K_+ \neq \emptyset$"> and
- denote by <!-- MATH
- $\Omega = \partial K_-\cup\partial K_+$
- -->
-<IMG
- WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img60.gif"
- ALT="$ \Omega = \partial K_-\cup\partial K_+$"> their union. A function <IMG
- WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img61.gif"
- ALT="$ v$"> is in <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"> if and only if the restricion <IMG
- WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img62.gif"
- ALT="$ v_-$"> of <IMG
- WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img61.gif"
- ALT="$ v$"> to <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$">
- is in <!-- MATH
- $H(\mathop{\rm curl}; K_-)$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img63.gif"
- ALT="$ H(\mathop{\rm curl}; K_-)$">, the restricion <IMG
- WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img64.gif"
- ALT="$ v_+$"> of <IMG
- WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img61.gif"
- ALT="$ v$"> to <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> is in <!-- MATH
- $H(\mathop{\rm curl}; K_+)$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img65.gif"
- ALT="$ H(\mathop{\rm curl}; K_+)$"> <I>and</I> the <I>tangential</I> jump over <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> vanishes: <!-- MATH
- $v_-\wedge n_- + v_+\wedge n_+ = 0$
- -->
-<IMG
- WIDTH="181" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img67.gif"
- ALT="$ v_-\wedge n_- + v_+\wedge n_+ = 0$"> on <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">.
- </DIV><P></P>
-<FONT SIZE="-1">
- </FONT><FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- The proposition follows from choosing an appropriate test function and integrating by parts (global and local).
- In order to <I>localise</I> the result of the Trace Theorem, we must choose a testfunction from the space <!-- MATH
- $H^{\frac{1}{2}}_{00}(e)$
- -->
-<IMG
- WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
- SRC="img68.gif"
- ALT="$ H^{\frac{1}{2}}_{00}(e)$">. These
- functions vanish at the endpoints of <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> and can therefore be extended by zero to a <!-- MATH
- $H^{\frac{1}{2}}(\partial \Omega )$
- -->
-<IMG
- WIDTH="69" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img69.gif"
- ALT="$ H^{\frac{1}{2}}(\partial \Omega )$">-function. From the comparison
- of local (on <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$"> and <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> separately) and global (on <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $">) integration by parts it followas then that the tangential jump vanishes in the
- dual space of <!-- MATH
- $H^{\frac{1}{2}}_{00}(e)$
- -->
-<IMG
- WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
- SRC="img68.gif"
- ALT="$ H^{\frac{1}{2}}_{00}(e)$">. By densitiy properties of <!-- MATH
- $H^{\frac{1}{2}}_{00}(e)$
- -->
-<IMG
- WIDTH="57" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
- SRC="img68.gif"
- ALT="$ H^{\frac{1}{2}}_{00}(e)$"> it follows that the tangential traces vanish in
- the "correct space" as well. The "correct space" would be <!-- MATH
- $H^{-\frac{1}{2}}(e)$
- -->
-<IMG
- WIDTH="65" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img70.gif"
- ALT="$ H^{-\frac{1}{2}}(e)$"> if we have no further regularity of <!-- MATH
- $\underline v_-$
- -->
-<IMG
- WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img71.gif"
- ALT="$ \underline v_-$"> and <!-- MATH
- $\underline v_+$
- -->
-<IMG
- WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img72.gif"
- ALT="$ \underline v_+$">, and it
- would be <IMG
- WIDTH="47" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img73.gif"
- ALT="$ L^2(e)$"> if <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$"> is elementwise in <IMG
- WIDTH="30" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img74.gif"
- ALT="$ H^1$"> (e. g. for piecewise polynomial <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$">).
-
-</FONT>
-<P>
-
-<H2><A NAME="SECTION00013000000000000000">
-1.3 Variational formulation of the model problem</A>
-</H2><FONT SIZE="-1">
-In the previous sections we introduced the space <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">, an integration-by-parts formula and the notion of trace for an
-<!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-function. In this framework, the variational formulation of the model problem (<A HREF="node1.html#eq:model_problem">1</A>) reads:
-</FONT><DL COMPACT>
-<DT></DT>
-<DD>Find <!-- MATH
- $\underline u \in H_0(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img75.gif"
- ALT="$ \underline u \in H_0(\mathop{\rm curl};\Omega )$"> such that for all test functions <!-- MATH
- $\underline v \in H_0(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="122" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img76.gif"
- ALT="$ \underline v \in H_0(\mathop{\rm curl};\Omega )$"> holds
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_varform"></A><!-- MATH
- \begin{equation}
-\int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx = \int_{\Omega } \underline f\, \cdot \underline v\,dx
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="378" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img77.gif"
- ALT="$\displaystyle \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\...
-..., \cdot \underline v\,dx = \int_{\Omega } \underline f\, \cdot \underline v\,dx$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(3)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-</DD>
-</DL><FONT SIZE="-1">
-With our assumptions on the data, the forms
-</FONT><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- a(\underline u,\underline v) &:= \int_{\Omega } \mathop{\rm curl}\underline u\, \mathop{\rm curl}\underline v\,dx + \int_{\Omega } c(x)\, \underline u\, \cdot \underline v\,dx \\
- l(\underline v) & := \int_{\Omega } \underline f\, \cdot \underline v\,dx
-\end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="454" HEIGHT="92" BORDER="0"
- SRC="img78.gif"
- ALT="\begin{displaymath}\begin{split}a(\underline u,\underline v) &:= \int_{\Omega } ...
-...int_{\Omega } \underline f\, \cdot \underline v\,dx \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-are continuous and the bilinear form <!-- MATH
- $a(\cdot,\cdot)$
- -->
-<IMG
- WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img79.gif"
- ALT="$ a(\cdot,\cdot)$"> is coercive on
-<!-- MATH
- $H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="198" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img80.gif"
- ALT="$ H_0(\mathop{\rm curl};\Omega )\times H_0(\mathop{\rm curl};\Omega )$">. By the Lax-Milgram lemma it follows, that there exists a unique solution
-<!-- MATH
- $\underline u \in H_0(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img75.gif"
- ALT="$ \underline u \in H_0(\mathop{\rm curl};\Omega )$"> of (<A HREF="node1.html#eq:_varform">3</A>).
-</FONT>
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
+++ /dev/null
-<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
-
-<!--Converted with LaTeX2HTML 2K.1beta (1.47)
-original version by: Nikos Drakos, CBLU, University of Leeds
-* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
-* with significant contributions from:
- Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
-<HTML>
-<HEAD>
-<TITLE>2 Nédélec's elements of first type for </TITLE>
-<META NAME="description" CONTENT="2 Nédélec's elements of first type for ">
-<META NAME="keywords" CONTENT="main">
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-</HEAD>
-
-<BODY >
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
-
-<UL>
-<LI><A NAME="tex2html59"
- HREF="#SECTION00021000000000000000">2.1 Construction of Nédélec elements on tetrahedral grids</A>
-<UL>
-<LI><A NAME="tex2html60"
- HREF="#SECTION00021100000000000000">2.1.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html61"
- HREF="#SECTION00021200000000000000">2.1.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html62"
- HREF="#SECTION00021300000000000000">2.1.3 Piola transformation</A>
-<LI><A NAME="tex2html63"
- HREF="#SECTION00021400000000000000">2.1.4 Transformation of the curl in 2d</A>
-<LI><A NAME="tex2html64"
- HREF="#SECTION00021500000000000000">2.1.5 Transformation of the curl in 3d</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html65"
- HREF="#SECTION00022000000000000000">2.2 Nédélec Elements on affine quadrilateral or hexahedral grids</A>
-<UL>
-<LI><A NAME="tex2html66"
- HREF="#SECTION00022100000000000000">2.2.1 Polynomial spaces on the reference element</A>
-<LI><A NAME="tex2html67"
- HREF="#SECTION00022200000000000000">2.2.2 Degrees of freedom on the reference element</A>
-<LI><A NAME="tex2html68"
- HREF="#SECTION00022300000000000000">2.2.3 Transformation of the vector field</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html69"
- HREF="#SECTION00023000000000000000">2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
-<UL>
-<LI><A NAME="tex2html70"
- HREF="#SECTION00023100000000000000">2.3.1 Bilinear elements in 2d</A>
-<LI><A NAME="tex2html71"
- HREF="#SECTION00023200000000000000">2.3.2 Trilinear elements in 3d</A>
-</UL>
-<BR>
-<LI><A NAME="tex2html72"
- HREF="#SECTION00024000000000000000">2.4 Construction of global shape functions</A>
-<LI><A NAME="tex2html73"
- HREF="#SECTION00025000000000000000">2.5 Approximation and convergence results</A>
-</UL>
-<!--End of Table of Child-Links-->
-<HR>
-
-<H1><A NAME="SECTION00020000000000000000">
-2 <FONT SIZE="+1">Nédélec's elements of first type for <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"></FONT></A>
-</H1><FONT SIZE="-1">
- In this section we will present present <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming vector-valued finite elements,
- the Nédélec elements of first type (cf. [<A
- HREF="node4.html#Ned1">8</A>]), which can be used to discretize the variational problem (<A HREF="node1.html#eq:_varform">3</A>).
-<BR></FONT>
-<P>
-<FONT SIZE="-1">In order to define a finite element we must specify
- </FONT><DL>
-<DT><STRONG>the geometry</STRONG></DT>
-<DD>We choose a reference element <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> and a change of variables <!-- MATH
- $F_K(\hat{x})$
- -->
-<IMG
- WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img82.gif"
- ALT="$ F_K(\hat{x})$">, the element map.
- We set <!-- MATH
- $K = F_K(\hat{K})$
- -->
-<IMG
- WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img83.gif"
- ALT="$ K = F_K(\hat{K})$">.
-
-</DD>
-<DT><STRONG>a function space</STRONG></DT>
-<DD>We need a <I>finite dimensional</I> function space <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img84.gif"
- ALT="$ \hat{R}$">, typically a space of polynomials, on the reference
- cell, plus a transformation of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img84.gif"
- ALT="$ \hat{R}$"> to a function space <IMG
- WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img85.gif"
- ALT="$ R_K$"> on a general cell <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$">.
-
-</DD>
-<DT><STRONG>dofs</STRONG></DT>
-<DD>We have to define a set of dofs <!-- MATH
- $\mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$
- -->
-<IMG
- WIDTH="119" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img87.gif"
- ALT="$ \mathcal{A} = \{\alpha_i(\cdot)\}_{i=1}^N$">. These are linear functionals on
- <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img84.gif"
- ALT="$ \hat{R}$"> and <!-- MATH
- $N < \infty$
- -->
-<IMG
- WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img88.gif"
- ALT="$ N < \infty$"> is the dimension of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img84.gif"
- ALT="$ \hat{R}$">. <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> should be <I>unisolvent</I>, that is, the dofs <!-- MATH
- $\alpha_i(\cdot)$
- -->
-<IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img90.gif"
- ALT="$ \alpha_i(\cdot)$">
- are linearly independent.
-
-</DD>
-</DL>
-<P>
-<FONT SIZE="-1">First, we observe that for a conforming discretization of (<A HREF="node1.html#eq:_varform">3</A>) we cannot take vector-valued finite elements
- that are build by taking the standard nodal finite element spaces of globally continuous functions for each vector component.
- For <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-functions, the only continuity condition is the continuity of the tangential component over cell boundaries.
- This fact will motivate the choice of appropriate degrees of freedom (abbreviated by dofs in the following).
-<BR>
-We will give an outline of the construction of the finite element spaces described by Nédélec in [<A
- HREF="node4.html#Ned1">8</A>]. In literature, they are also referred to as
- <I>Nédélec's elements of first type</I>.
-<BR></FONT>
-<P>
-
-<H2><A NAME="SECTION00021000000000000000">
-2.1 Construction of Nédélec elements on tetrahedral grids</A>
-</H2>
-<P>
-<FONT SIZE="-1">In this section, we denote by <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> the standard triangular or tetrahedral reference element.
-</FONT>
-<P>
-
-<H3><A NAME="SECTION00021100000000000000">
-2.1.1 Polynomial spaces on the reference element</A>
-</H3><FONT SIZE="-1">
- In [<A
- HREF="node4.html#Ned1">8</A>], Nédélec introduces the function spaces <!-- MATH
- $\hat{R} = \mathcal{R}^k$
- -->
-<IMG
- WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img91.gif"
- ALT="$ \hat{R} = \mathcal{R}^k$">, on which his finite element will be based. These spaces are subject to this
- section.
- For more details, consult [<A
- HREF="node4.html#Ned1">8</A>].
-<BR>
-We denote by <!-- MATH
- $\mathbb{P}_k(\hat{\Sigma})$
- -->
-<IMG
- WIDTH="51" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img92.gif"
- ALT="$ \mathbb{P}_k(\hat{\Sigma})$"> the space of polynomials of degree <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> on <!-- MATH
- $\hat{\Sigma}$
- -->
-<IMG
- WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img94.gif"
- ALT="$ \hat{\Sigma}$">, where <!-- MATH
- $\hat{\Sigma}$
- -->
-<IMG
- WIDTH="19" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img94.gif"
- ALT="$ \hat{\Sigma}$"> is an edge, a
- face of or the reference element itself.. The space <!-- MATH
- $\tilde{\mathbb{P}}_k$
- -->
-<IMG
- WIDTH="25" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img95.gif"
- ALT="$ \tilde{\mathbb{P}}_k$"> of homogeneous
- polynomials of degree <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> is the span of monomials of total degree <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> in <IMG
- WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img96.gif"
- ALT="$ d$"> variables on <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">.
-</FONT>
-<P>
-<P>
-<DIV><B>D<SMALL>EFINITION</SMALL> 3</B>
-We define the auxiliary space
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="def:_space_Sk"></A><!-- MATH
- \begin{equation}
-\mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d : \underline p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="317" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
- SRC="img97.gif"
- ALT="$\displaystyle \mathcal{S}^k := \{\, \underline p \in (\tilde{\mathbb{P}}_k)^d :...
-...ine p \cdot \hat{\underline x} = \sum_{i=1}^{d} p_i\,\hat{x}_i \equiv 0 \,\}\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(4)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-with <!-- MATH
- $\hat{x} \in \hat{K}$
- -->
-<IMG
- WIDTH="54" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img98.gif"
- ALT="$ \hat{x} \in \hat{K}$">.
-<BR>
-The dimension of this space is <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> in the case <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$"> and <IMG
- WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img99.gif"
- ALT="$ k(k+2)$"> for <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">Nédélec's first family of <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite elements is based on the polynomial spaces
- </FONT><P>
-<DIV><B>D<SMALL>EFINITION</SMALL> 4</B>
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="def:_space_Rk"></A><!-- MATH
- \begin{equation}
-\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="196" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img100.gif"
- ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^d \oplus \mathcal{S}^k\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(5)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-These spaces have dimension
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- \mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \textrm{for} \quad d=2\,, \\
- \mathrm{dim} (\mathcal{R}^k) &= \frac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,.
- \end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="446" HEIGHT="72" BORDER="0"
- SRC="img101.gif"
- ALT="\begin{displaymath}\begin{split}\mathrm{dim} (\mathcal{R}^k) &= k(k+2) \qquad \t...
-...ac{(k+3)(k+2)k}{2} \qquad \textrm{for} \quad d=3\,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">In the two-dimensional case, an equivalent characterization of the space <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> is
-</FONT>
-<P>
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_spaces_2d"></A><!-- MATH
- \begin{equation}
-\mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1
- \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="293" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img103.gif"
- ALT="$\displaystyle \mathcal{R}^k = \left(\mathbb{P}_{k-1}(\hat{K}) \right)^2 \oplus ...
-..._{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(6)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- This can be seen by noting that for <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$">
- </FONT><!-- MATH
- \begin{displaymath}
-\tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \subseteq \mathcal{S}^k
- %%\left\{\v p\quad \big|\quad \v p = \tilde{p} \left(\begin{array}{cc} x_2 \\-x_1 \end{array}\right)\,,\, \tilde{p}
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="164" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img104.gif"
- ALT="$\displaystyle \tilde{\mathbb{P}}_{k-1}\, \left(\begin{array}{cc} \hat{x}_2 \\ ...
-...ilde{p} \left(\begin{array}{cc} x_2 \\ -x_1 \end{array}\right)\,,\, \tilde{p}
-$">
-</DIV><P></P><FONT SIZE="-1">
- obviously holds.
- Moreover, the dimension of the space <!-- MATH
- $\tilde{\mathbb{P}}_{k-1}$
- -->
-<IMG
- WIDTH="43" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img105.gif"
- ALT="$ \tilde{\mathbb{P}}_{k-1}$"> of homogeneous polynomials of degree <IMG
- WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img106.gif"
- ALT="$ k-1$"> in two variables is <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> and this is
- also the dimension <!-- MATH
- $\mathcal{S}^k$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img107.gif"
- ALT="$ \mathcal{S}^k$">. This proves the stated equivalent representation of the space <!-- MATH
- $\mathcal{S}^k$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img107.gif"
- ALT="$ \mathcal{S}^k$">.
-<BR>
-<BR>
-We illustrate these definitions with some examples. We start with the case <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img13.gif"
- ALT="$ d=2$"> and consider the spaces of polynomials of degree
- <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$"> and <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img109.gif"
- ALT="$ k=2$">:
-</FONT>
-<P>
-<P>
-<DIV><B>E<SMALL>XAMPLE</SMALL> 1</B>
-
-<P>
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_R1"></A><!-- MATH
- \begin{equation}
-\mathcal{R}^1 = \left\langle
- \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,_,
- \left(\begin{array}{cc} 0 \\1 \end{array}\right)\,_,
- \left(\begin{array}{cc} \hat{x}_2 \\-\hat{x}_1 \end{array}\right) \right\rangle
-
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="278" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img110.gif"
- ALT="$\displaystyle \mathcal{R}^1 = \left\langle \left(\begin{array}{cc} 1 \\ 0 \end...
-...eft(\begin{array}{cc} \hat{x}_2 \\ -\hat{x}_1 \end{array}\right) \right\rangle$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(7)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-<!-- MATH
- \begin{displaymath}
-\mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
- \left\langle
- \left(\begin{array}{cc} \hat{x}_1\,\hat{x}_2 \\-{\hat{x}_1}^2 \end{array}\right)\,_,
- \left(\begin{array}{cc} {\hat{x}_2}^2 \\-\hat{x}_1\,\hat{x}_2 \end{array}\right)
- \right\rangle
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img111.gif"
- ALT="$\displaystyle \mathcal{R}^2 = \left(\mathbb{P}_{1}(\hat{K}) \right)^2 \oplus
-\...
-...{cc} {\hat{x}_2}^2 \\ -\hat{x}_1\,\hat{x}_2 \end{array}\right)
-\right\rangle
-$">
-</DIV><P></P>
-
-<P>
-</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">To illustrate a case for <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img27.gif"
- ALT="$ d=3$">, we consider the lowest polynomial degree <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$">:
- </FONT><P>
-<DIV><B>E<SMALL>XAMPLE</SMALL> 2</B>
-We have to specify a basis for <!-- MATH
- $\mathcal{S}^1$
- -->
-<IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img112.gif"
- ALT="$ \mathcal{S}^1$">:
-<BR>
-Let <!-- MATH
- $\underline p$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img113.gif"
- ALT="$ \underline p$"> be a polynomial in <!-- MATH
- $(\mathbb{P}_{1}(\hat{K}))^3$
- -->
-<IMG
- WIDTH="75" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img114.gif"
- ALT="$ (\mathbb{P}_{1}(\hat{K}))^3$"> with componentwise representation
- <!-- MATH
- \begin{displaymath}
-p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="234" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
- SRC="img115.gif"
- ALT="$\displaystyle p_i = \sum_{j=1}^3 a_{ij} \hat{x}_j\,, \qquad i=1,2,3\,.
-$">
-</DIV><P></P>
-The condition for <!-- MATH
- $\underline p$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img113.gif"
- ALT="$ \underline p$"> being in <!-- MATH
- $\mathcal{S}^1$
- -->
-<IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img112.gif"
- ALT="$ \mathcal{S}^1$"> is
- <!-- MATH
- \begin{displaymath}
-\underline p \cdot \hat{\underline x} =
- \sum_{i=1}^3 a_{ii}\hat{x}_i^2 + \sum_{\substack{i,j=1 \\j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="327" HEIGHT="86" ALIGN="MIDDLE" BORDER="0"
- SRC="img116.gif"
- ALT="$\displaystyle \underline p \cdot \hat{\underline x} =
-\sum_{i=1}^3 a_{ii}\hat{...
-...um_{\substack{i,j=1 \\ j>i}}^3 (a_{ij}+a_{ji})\hat{x}_i \hat{x}_j \equiv 0\,.
-$">
-</DIV><P></P>
-This leads to the condition on the coefficients of a polynomial in <!-- MATH
- $\mathcal{S}^1$
- -->
-<IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img112.gif"
- ALT="$ \mathcal{S}^1$">:
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- &a_{11}=a_{22}=a_{33} = 0 \\
- &a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,.
- \end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="439" HEIGHT="55" BORDER="0"
- SRC="img117.gif"
- ALT="\begin{displaymath}\begin{split}&a_{11}=a_{22}=a_{33} = 0 \\ &a_{12}= - a_{21}\,,\quad a_{13}= - a_{31}\,,\quad a_{23}= - a_{32}\,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-With the basis of <!-- MATH
- $\mathcal{S}^1$
- -->
-<IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img112.gif"
- ALT="$ \mathcal{S}^1$"> which is obtained by choosing <!-- MATH
- $a_{ij} = 1$
- -->
-<IMG
- WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img118.gif"
- ALT="$ a_{ij} = 1$">, <IMG
- WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img119.gif"
- ALT="$ i=1,2,3$">, <IMG
- WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img120.gif"
- ALT="$ j>i$"> and setting all the other
- coefficients to zero, we get
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus
- \left\langle
- \left(\begin{array}{ccc} 0 \\\hat{x}_3 \\\hat{x}_2 \end{array}\right)\,_,\,
- \left(\begin{array}{ccc} \hat{x}_3 \\0 \\\hat{x}_1 \end{array}\right)\,_,\,
- \left(\begin{array}{ccc} \hat{x}_2 \\\hat{x}_1 \\0 \end{array}\right)
- \right\rangle
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="402" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
- SRC="img121.gif"
- ALT="$\displaystyle \mathcal{R}^1 = \left(\mathbb{P}_{0}(\hat{K}) \right)^3 \oplus \l...
-...egin{array}{ccc} \hat{x}_2 \\ \hat{x}_1 \\ 0 \end{array}\right) \right\rangle$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">We remark at this point that the spaces <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> do not span the whole <!-- MATH
- $(\mathbb{P}_{k}(\hat{K}) )^d$
- -->
-<IMG
- WIDTH="76" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img122.gif"
- ALT="$ (\mathbb{P}_{k}(\hat{K}) )^d$">.
- An <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM based on full polynomial spaces, the so called
- <I>Nédélec elements of second type</I>, was introduced in 1986 by Nédélec in [<A
- HREF="node4.html#Ned2">10</A>].
-</FONT>
-<P>
-<P>
-<DIV><B>R<SMALL>EMARK</SMALL> 3</B>
-The original, rather technical, representation of the spaces <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> is given in Definition 2 in [<A
- HREF="node4.html#Ned1">8</A>].
- Nédélec uses this representation in most of his proofs. We will not refer to it here.
- </DIV><P></P>
-
-<P>
-
-<H3><A NAME="SECTION00021200000000000000"></A>
- <A NAME="subsubsect:_dofs_on_tria"></A>
-<BR>
-2.1.2 Degrees of freedom on the reference element
-</H3><FONT SIZE="-1">
- In this section we define the set <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> of dofs, which is a set of linear functionals on <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$">.
- </FONT><P>
-<DIV><B>R<SMALL>EMARK</SMALL> 4</B>
-Recall that the dimension of the spaces of polynomials of degree <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> in <IMG
- WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img123.gif"
- ALT="$ n$"> variables is <!-- MATH
- $n+k+2 \choose n$
- -->
-<IMG
- WIDTH="66" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img124.gif"
- ALT="$ n+k+2 \choose n$">.
- </DIV><P></P>
-
-<P>
-<P>
-<DIV><A NAME="def:_dofs_2d"><B>D<SMALL>EFINITION</SMALL> 5</B></A>
-Let <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> be the reference triangle and <!-- MATH
- $\hat{\underline t}$
- -->
-<IMG
- WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img125.gif"
- ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>.
- The set of degrees of freedom <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> on <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> in the 2d case consists of the linear functionals
- <DL>
-<DT><STRONG>edge dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall \hat{\varphi }
- \in \mathbb{P}_{k-1}(\hat{e})\,,
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img126.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
-...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> of <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">. We have a total of <IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img128.gif"
- ALT="$ 3k$"> of edge dofs.
-
-<P>
-</DD>
-<DT><STRONG>inner dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad \forall \hat{\underline \varphi }
- \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img129.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
-...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-2}(\hat{K}))^2\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-We have a total of <IMG
- WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img130.gif"
- ALT="$ k(k-1)$"> of inner dofs.
-
-<P>
-</DD>
-</DL></DIV><P></P>
-
-<P>
-<P>
-<DIV><A NAME="def:_dofs_3d"><B>D<SMALL>EFINITION</SMALL> 6</B></A>
-Let <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> be the reference tetrahedron, <!-- MATH
- $\hat{\underline t}$
- -->
-<IMG
- WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img125.gif"
- ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A>
- and <!-- MATH
- $\hat{\underline n}$
- -->
-<IMG
- WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img131.gif"
- ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face.
- The set of degrees of freedom <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> on <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> in the 3d case consists of the linear functionals
- <DL>
-<DT><STRONG>edge dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s} \quad \forall
- \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="288" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img126.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
-...arphi }\,d\hat{s} \quad \forall \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> of the tetrahedron <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">. We have a total of <IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img132.gif"
- ALT="$ 6k$"> of edge dofs.
-
-<P>
-</DD>
-<DT><STRONG>face dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \quad
- \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="335" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img133.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
-... }\,d\hat{a} \quad \forall \hat{\varphi } \in (\mathbb{P}_{k-2}(\hat{f}) )^2\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every face <IMG
- WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img134.gif"
- ALT="$ \hat{f}$"> of the tetrahedron <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">. We have a total of <IMG
- WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img135.gif"
- ALT="$ 4k(k-1)$"> of face dofs.
-
-<P>
-</DD>
-<DT><STRONG>inner dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \quad
- \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="305" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img136.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{K}} \hat{\underlin...
-...x} \quad \forall \hat{\underline \varphi } \in (\mathbb{P}_{k-3}(\hat{K}))^3\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-We have a total of <!-- MATH
- $\frac{k(k-1)(k-2)}{2}$
- -->
-<IMG
- WIDTH="88" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img137.gif"
- ALT="$ \frac{k(k-1)(k-2)}{2}$"> of inner dofs.
-
-</DD>
-</DL></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">We note that in the case of lowest order elements, i. e. <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$">, only edge dofs occur. This is not so for higher order elements. For
- <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img109.gif"
- ALT="$ k=2$"> we additionally have inner dofs in the 2d case and face dofs in the 3d case. For <IMG
- WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img138.gif"
- ALT="$ k\leq3$"> we have all types of dofs in both
- cases.
-<BR>
-We also note that the total number of dofs equals the dimension of the spaces <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$">, as it should be.
-<BR>
-The representation of the <I>interface</I> dofs, that is edge dofs in 2d, edge and face dofs in 3d, is motivated by the continuity
- condition on <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-functions stated in proposition <A HREF="node1.html#prop:_no_jumps">1</A>.
-</FONT>
-<P>
-<P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 2</B>
-The set <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> of dofs befined above is unisolvent on <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$">.
- <!-- MATH
- $\hat{\underline u}\in \mathcal{R}^k$
- -->
-<IMG
- WIDTH="61" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img139.gif"
- ALT="$ \hat{\underline u}\in \mathcal{R}^k$"> is uniquely defined by the moments <!-- MATH
- $\hat{\alpha}(\hat{\underline u})$
- -->
-<IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img140.gif"
- ALT="$ \hat{\alpha}(\hat{\underline u})$">.
- </DIV><P></P>
-<FONT SIZE="-1">
- </FONT><FONT SIZE="-1">P<SMALL>ROOF. </SMALL>See [<A
- HREF="node4.html#Ned1">8</A>], proof of theorem 1 and preceeding lemmas.
-</FONT>
-<P>
-<P>
-<DIV><A NAME="ex:_shape_functions"><B>E<SMALL>XAMPLE</SMALL> 3</B></A> (Reference shape functions of lowest order for Nédélec elements on triangular meshes)
-Let the reference element be the triangle
- <!-- MATH
- $\hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$
- -->
-<IMG
- WIDTH="374" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img141.gif"
- ALT="$ \hat{K} = \left\{ (\hat{x},\hat{y})\in\mathbb{R}^2:\quad 0\leq \hat{x}\leq 1\,,\,\, 0\leq \hat{y}\leq 1-\hat{x} \right\}$">.
- Label the edges couterclockwise startung with <!-- MATH
- $\hat{e}_0 = \overline{(0,0),(1,0)}$
- -->
-<IMG
- WIDTH="131" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img142.gif"
- ALT="$ \hat{e}_0 = \overline{(0,0),(1,0)}$">.
- The tangential vectors to the edges are (oriented counterclockwise)
- <!-- MATH
- \begin{displaymath}
-\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
- \hat{\underline t}_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} -1 \\1 \end{array}\right)\,,\quad
- \hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="398" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img143.gif"
- ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array...
-...
-\hat{\underline t}_2 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,.
-$">
-</DIV><P></P>
-The underlying function space for lowest order Nédélec elements on a triangular mesh is <!-- MATH
- $\mathcal{R}^1$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img144.gif"
- ALT="$ \mathcal{R}^1$"> from (<A HREF="#eq:_R1"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>).
-<BR>
-In the case of <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$"> only egde-dofs occur. On <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> we have dofs of the type
- <!-- MATH
- $\int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$
- -->
-<IMG
- WIDTH="202" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img145.gif"
- ALT="$ \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \forall \hat{\varphi } \in \mathbb{P}_{0}(\hat{e}_i)$">.
- More precisely, since <!-- MATH
- $\varphi \equiv 1$
- -->
-<IMG
- WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img146.gif"
- ALT="$ \varphi \equiv 1$"> is a basis for <!-- MATH
- $\mathbb{P}_{0}(\hat{e}_i)$
- -->
-<IMG
- WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img147.gif"
- ALT="$ \mathbb{P}_{0}(\hat{e}_i)$"> we have the three dofs
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="250" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img148.gif"
- ALT="$\displaystyle \hat{\alpha}_i(\hat{\underline u}) = \int_{\hat{e}_i} (\hat{\underline t}\cdot \hat{\underline u})\,d\hat{s} \quad i=0,1,2\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-In order to construct a FE-basis <!-- MATH
- $\hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$
- -->
-<IMG
- WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img149.gif"
- ALT="$ \hat{\underline N}_0,\hat{\underline N}_1,\hat{\underline N}_2$"> for <!-- MATH
- $\mathcal{R}^1$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img144.gif"
- ALT="$ \mathcal{R}^1$"> with respect to these dofs, we require
- <!-- MATH
- $\hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$
- -->
-<IMG
- WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img150.gif"
- ALT="$ \hat{\alpha}_i(\hat{\underline N}_j) = \delta_{ij}$">. This leads to a linear system for
- the coefficients of the <!-- MATH
- $\hat{\underline N}_i$
- -->
-<IMG
- WIDTH="28" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img151.gif"
- ALT="$ \hat{\underline N}_i$"> in a general basis of <!-- MATH
- $\mathcal{R}^1$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img144.gif"
- ALT="$ \mathcal{R}^1$">. In the case of lowest order elements, it is easy to
- verify that we have
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation}
-\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\\hat{x} \end{array}\right)\,,\quad
- \hat{\underline N}_1 = \left(\begin{array}{cc} -\hat{y} \\\hat{x} \end{array}\right)\,,\quad
- \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\\hat{x}-1 \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="446" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img152.gif"
- ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ \hat...
-...ine N}_2 = \left(\begin{array}{cc} -\hat{y} \\ \hat{x}-1 \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(8)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-
-<H3><A NAME="SECTION00021300000000000000"></A> <A NAME="sec:_Piola_for_triangles"></A>
-<BR>
-2.1.3 Piola transformation
-</H3><FONT SIZE="-1">
- An affine triangle or tetrahedron <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> is described by the affine element map
- </FONT><!-- MATH
- \begin{displaymath}
-K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img153.gif"
- ALT="$\displaystyle K \ni x = F_K(\hat{x}) = B_K \hat{x} + b_K
-$">
-</DIV><P></P>
-<P>
-<FONT SIZE="-1">In standard <!-- MATH
- $H^1(\Omega )$
- -->
-<IMG
- WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img37.gif"
- ALT="$ H^1(\Omega )$">-conforming FEM, the shape functions <IMG
- WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img154.gif"
- ALT="$ N_i$"> on a general cell <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> are obtained from the reference shape functions
- <IMG
- WIDTH="26" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img155.gif"
- ALT="$ \hat{N}_i$"> on the reference element <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> by the pull-back
- </FONT><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="187" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
- SRC="img156.gif"
- ALT="$\displaystyle N_i(x) = \left( \hat{N}_i \circ F_K^{-1} \right)(x)$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- In the case of <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming Nédélec FEM we cannot transforme our shape function in this way. The pull-back of a
- <!-- MATH
- $H(\mathop{\rm curl};\hat{K})$
- -->
-<IMG
- WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img157.gif"
- ALT="$ H(\mathop{\rm curl};\hat{K})$">-function needs not to be in <!-- MATH
- $H(\mathop{\rm curl}; K)$
- -->
-<IMG
- WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img158.gif"
- ALT="$ H(\mathop{\rm curl}; K)$">. In addition, the pull-back is not an <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$">-isomorphism and it
- does not lead to an <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method if prescribing the dofs by definitions <A HREF="node2.html#def:_dofs_2d">5</A> or <A HREF="#def:_dofs_____3d"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>.
-<BR>
-In Nédélec's FEM (or, more general, in <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM), the shape functions are transformed by the following covariant
- transformation for vector-fields:
-</FONT>
-<P>
-<DL COMPACT>
-<DT></DT>
-<DD>The element shape functions <!-- MATH
- $\underline N_i(x)$
- -->
-<IMG
- WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img159.gif"
- ALT="$ \underline N_i(x)$"> on the element <!-- MATH
- $K = F_K(\hat{K})$
- -->
-<IMG
- WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img83.gif"
- ALT="$ K = F_K(\hat{K})$"> are obtained
- from the reference shape functions by
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Piola"></A><!-- MATH
- \begin{equation}
-\underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="329" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
- SRC="img160.gif"
- ALT="$\displaystyle \underline N_i(x) = \mathcal{P}_K (\hat{\underline N}_i) = \left(\hat{D}F_K^{-T} \hat{\underline N}_i\right) \circ F_K^{-1} (x)\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(9)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-where <!-- MATH
- $\hat{D}F_K$
- -->
-<IMG
- WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img161.gif"
- ALT="$ \hat{D}F_K$"> is the jacobian <!-- MATH
- $\frac{d}{d\hat{x}}F_K(\hat{x})$
- -->
-<IMG
- WIDTH="73" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img162.gif"
- ALT="$ \frac{d}{d\hat{x}}F_K(\hat{x})$"> of the element map.
-
-</DD>
-</DL>
-<P>
-<FONT SIZE="-1">In literature, an equivalent to this transformation for <!-- MATH
- $H(\mathop{\rm div}; \Omega )$
- -->
-<IMG
- WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img163.gif"
- ALT="$ H(\mathop{\rm div}; \Omega )$">-conforming FEM (which in that case is a contravariant map)
- is referred to as <I>Piola transformation</I>, cf. [<A
- HREF="node4.html#Brezzi-Fortin">3</A>] pp. 97.
-<BR>
-Here, we will refer to the transformation (<A HREF="node2.html#eq:_Piola">9</A>) of the vector field also as <I>Piola transformation</I>.
-<BR>
-We note that the gradients of scalar nodal <!-- MATH
- $H^1(\Omega )$
- -->
-<IMG
- WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img37.gif"
- ALT="$ H^1(\Omega )$">-conforming finite elements transform according to the Piola
- transformation (<A HREF="node2.html#eq:_Piola">9</A>).
-</FONT>
-<P>
-<FONT SIZE="-1">In the case of tetrahedral elements and affine element map <!-- MATH
- $F_K(\hat{x}) = B_K \hat{x} + b_k$
- -->
-<IMG
- WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img164.gif"
- ALT="$ F_K(\hat{x}) = B_K \hat{x} + b_k$">, the jacobian <!-- MATH
- $\hat{D}F_K$
- -->
-<IMG
- WIDTH="46" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img161.gif"
- ALT="$ \hat{D}F_K$">
- is just the constant matrix <IMG
- WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img165.gif"
- ALT="$ B_K$"> and we have
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Piola_tri"></A><!-- MATH
- \begin{equation}
-\underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="279" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img166.gif"
- ALT="$\displaystyle \underline v(x) = \mathcal{P}_K (\hat{\underline v}) = B_K^{-T} \left(\hat{\underline v} \circ F_K^{-1} \right)(x)\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(10)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-
-<H3><A NAME="SECTION00021400000000000000"></A>
- <A NAME="subsubsect:_2d-curl_on_tria"></A>
-<BR>
-2.1.4 Transformation of the curl in 2d
-</H3><FONT SIZE="-1">
- For <!-- MATH
- $\Omega \subset\mathbb{R}^2$
- -->
-<IMG
- WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img167.gif"
- ALT="$ \Omega \subset\mathbb{R}^2$">, we noted in remark <A HREF="node1.html#re:_rotation">1</A> that vector fields in <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$"> can be represented as
- rotated <!-- MATH
- $H(\mathop{\rm div}; \Omega )$
- -->
-<IMG
- WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img163.gif"
- ALT="$ H(\mathop{\rm div}; \Omega )$"> vector fields.
- Moreover, it is easy to verify that
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_rot._jacobian"></A><!-- MATH
- \begin{equation}
-B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img168.gif"
- ALT="$\displaystyle B_K^{-T} = \det B_K^{-1}\,R^T B_K\,R \,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(11)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- where <IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img169.gif"
- ALT="$ R$"> is the rotation matrix from remark <A HREF="node1.html#re:_rotation">1</A>.
- Therefore, the properties of the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) in the 2d case can be derived directly
- from the properties
- of the <!-- MATH
- $H(\mathop{\rm div}; \Omega )$
- -->
-<IMG
- WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img163.gif"
- ALT="$ H(\mathop{\rm div}; \Omega )$">-Piola transformation stated in [<A
- HREF="node4.html#Brezzi-Fortin">3</A>] pp. 97.
-</FONT>
-<P>
-<P>
-<DIV><A NAME="th:_piola_2d"><B>T<SMALL>HEOREM</SMALL> 4</B></A> (Some properties of 2d Piola transformation for affine element map)
-Let <!-- MATH
- $\underline v(x) = \mathcal{P}_K(\hat{\underline v})$
- -->
-<IMG
- WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img170.gif"
- ALT="$ \underline v(x) = \mathcal{P}_K(\hat{\underline v})$">, <!-- MATH
- $\varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$
- -->
-<IMG
- WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img171.gif"
- ALT="$ \varphi (x) = \left( \hat{\varphi }\circ F_K^{-1} \right)(x)$">, <!-- MATH
- $\hat{x} =
-F_K^{-1}(x)$
- -->
-<IMG
- WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img172.gif"
- ALT="$ \hat{x} =
-F_K^{-1}(x)$">, with affine element map <IMG
- WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img173.gif"
- ALT="$ F_K$">.
- <DL COMPACT>
-<DT>(i)</DT>
-<DD>The gradient <!-- MATH
- $D\underline v$
- -->
-<IMG
- WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img174.gif"
- ALT="$ D\underline v$"> transforms according to
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_gradient_trafo_2d"></A><!-- MATH
- \begin{equation}
-D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img175.gif"
- ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(12)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-
-</DD>
-<DT>(ii)</DT>
-<DD>The curl transforms according to
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:curl_trafo_2d"></A><!-- MATH
- \begin{equation}
-\mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="176" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img176.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \det B_K^{-1} \widehat{\mathop{\rm curl}} \hat{\underline v}\, .$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(13)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-As a consequence we see that <!-- MATH
- $H(\mathop{\rm curl};K)$
- -->
-<IMG
- WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img158.gif"
- ALT="$ H(\mathop{\rm curl}; K)$"> is isomorphic to <!-- MATH
- $H(\mathop{\rm curl};\hat{K})$
- -->
-<IMG
- WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img157.gif"
- ALT="$ H(\mathop{\rm curl};\hat{K})$"> under the Piola transformation
- (<A HREF="node2.html#eq:_Piola_tri">10</A>).
-
-</DD>
-</DL></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- </FONT><DL COMPACT>
-<DT>(i)</DT>
-<DD>Chain rule
-
-</DD>
-<DT>(ii)</DT>
-<DD>We use that the 2d <!-- MATH
- $\mathop{\rm curl}$
- -->
-<IMG
- WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img19.gif"
- ALT="$ \mathop{\rm curl}$"> operator is just the trace of the rotated jacobian <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img177.gif"
- ALT="$ R\,Dv$">. By remark <A HREF="node2.html#eq:_rot._jacobian">11</A>,
- we can replace <IMG
- WIDTH="42" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img178.gif"
- ALT="$ B_K^{-T}$"> and we get that <IMG
- WIDTH="47" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img177.gif"
- ALT="$ R\,Dv$"> is affine-equivalent to <!-- MATH
- $\det B_K^{-1}\,R\,\hat{D}\hat{v}$
- -->
-<IMG
- WIDTH="109" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img179.gif"
- ALT="$ \det B_K^{-1}\,R\,\hat{D}\hat{v}$">,
- which proves (ii).
-
-</DD>
-</DL><FONT SIZE="-1">
-
-</FONT>
-<P>
-<P>
-<DIV><B>C<SMALL>OROLLARY</SMALL> 1</B>
-From (ii) in theorem <A HREF="node2.html#th:_piola_2d">4</A> we deduce
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="240" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img180.gif"
- ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \varphi \, dx = \int_{\h...
-...}} \widehat{\mathop{\rm curl}}\hat{\underline v}\, \hat{\varphi } \,d\hat{x}\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-and we have, together with (ii) from theorem <A HREF="node2.html#th:_piola_2d">4</A>
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underline u \, dx = | B_K |^{-1}\,\int_{\hat{K}} \widehat{\mathop{\rm curl}}\hat{\underline v}\,
- \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="362" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img181.gif"
- ALT="$\displaystyle \int_{K} \mathop{\rm curl}\underline v\, \mathop{\rm curl}\underl...
-...hat{\underline v}\, \widehat{\mathop{\rm curl}}\hat{\underline u} \,d\hat{x}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1"></FONT>
-<P>
-
-<H3><A NAME="SECTION00021500000000000000"></A>
-<A NAME="subsubsect:_3d-curl_on_tria"></A>
-<BR>
-2.1.5 Transformation of the curl in 3d
-</H3><FONT SIZE="-1">
- In three dimensions, we cannot identify the curl-operator with the rotated
- gradient or with the divergence of a rotated vector field. We cannot, as in 2d, derive a transformation formula for the curl from the
- transformatin formula of the divergence.
-<BR>
-By the chain rule, we obtain the transformation of the gradient of a vector field <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$">, defined by the Piola transformation (<A HREF="#eq:_Piola_tri"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="crossref.gif"></A>) of a reference field <!-- MATH
- $\hat{\underline v}$
- -->
-<IMG
- WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img182.gif"
- ALT="$ \hat{\underline v}$">:
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_trafo_grad_3d"></A><!-- MATH
- \begin{equation}
-D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="159" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img175.gif"
- ALT="$\displaystyle D\underline v = B_K^{-T}\, \hat{D}\hat{\underline v}\, B_K^{-1}\, .$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(14)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-<FONT SIZE="-1">We introduce the skew symmetric matrix <!-- MATH
- $\mathop{\rm Curl}v$
- -->
-<IMG
- WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img183.gif"
- ALT="$ \mathop{\rm Curl}v$"> as
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Curl"></A><!-- MATH
- \begin{equation}
-\left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="183" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
- SRC="img184.gif"
- ALT="$\displaystyle \left({\mathop{\rm Curl}v}\right)_{ij} = \frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(15)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-<FONT SIZE="-1">We see that <!-- MATH
- $\mathop{\rm Curl}v = D\underline v^T - D\underline v$
- -->
-<IMG
- WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img185.gif"
- ALT="$ \mathop{\rm Curl}v = D\underline v^T - D\underline v$"> and therefore by (<A HREF="node2.html#eq:_trafo_grad_3d">14</A>)
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Curl_trafo"></A><!-- MATH
- \begin{equation}
-\mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="192" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img186.gif"
- ALT="$\displaystyle \mathop{\rm Curl}v = B_K^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,B_K^{-1}$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(16)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-<P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 3</B> (Transformation of the curl in 3d)
-Let <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> be the reference tetrahedron and <!-- MATH
- $K=F_K(\hat{K})$
- -->
-<IMG
- WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img83.gif"
- ALT="$ K = F_K(\hat{K})$"> an affine image of it.
- The curl of a vector field <!-- MATH
- $\underline v(x)$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img187.gif"
- ALT="$ \underline v(x)$"> on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$">, defined by the Piola transformation of a reference field <!-- MATH
- $\hat{\underline v}(\hat{x})$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img188.gif"
- ALT="$ \hat{\underline v}(\hat{x})$">
- transforms according to
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_curl_trafo_3d"></A><!-- MATH
- \begin{equation}
-\left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="307" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img189.gif"
- ALT="$\displaystyle \left(\mathop{\rm curl}\underline v\right)_i(x) = \det \mathrm{M_i}(x) \,, \qquad i= 1,2,3\,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(17)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-We obtain the matrix <!-- MATH
- $\mathrm{M_i}$
- -->
-<IMG
- WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img190.gif"
- ALT="$ \mathrm{M_i}$"> by replacing i-th column of the (constant) jacobian <!-- MATH
- $D(F_K^{-1}) = B_K^{-1}$
- -->
-<IMG
- WIDTH="122" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img191.gif"
- ALT="$ D(F_K^{-1}) = B_K^{-1}$">
- by the vector <!-- MATH
- $(\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$
- -->
-<IMG
- WIDTH="133" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img192.gif"
- ALT="$ (\widehat{\mathop{\rm curl}}\,\hat{\underline v}\circ F_K^{-1})(x)$">:
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll}
- (\widehat{\mathop{\rm curl}}\,\hat{v} \circ F_K^{-1})_k (x) & \textrm{if} \quad l=i \\
- (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i
- \end{array} \right.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="351" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
- SRC="img193.gif"
- ALT="$\displaystyle \left(\mathrm{M_i}\right)_{kl}(x) := \left\{ \begin{array}{ll} (\...
-...} \quad l=i \\ (B_K^{-1})_{kl} & \textrm{if} \quad l\neq i \end{array} \right.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
- (Note: an alternative, equivalent, transformation formula for the curl in 3d is given in proposition <A HREF="node2.html#prop:_Demko_curl">4</A>).
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- It holds
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_curl2Curl"></A><!-- MATH
- \begin{equation}
-\mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\mathop{\rm Curl}v})_{23} \\({\mathop{\rm Curl}v})_{31} \\({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="202" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
- SRC="img194.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \left( \begin{array}{ccc} ({\math...
-...\mathop{\rm Curl}v})_{31} \\ ({\mathop{\rm Curl}v})_{12} \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(18)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- We demonstrate the statement of the proposition for the first component of the curl, which is <!-- MATH
- $(\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$
- -->
-<IMG
- WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img195.gif"
- ALT="$ (\mathop{\rm curl}\underline v)_1 = {\mathop{\rm Curl}v}_{23}$">.
- Using the transformation (<A HREF="node2.html#eq:_Curl_trafo">16</A>), implicit summation over equal indices and the
- abbreviation <!-- MATH
- $b_{ij} := (B_K^{-1})_{ij}$
- -->
-<IMG
- WIDTH="111" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img196.gif"
- ALT="$ b_{ij} := (B_K^{-1})_{ij}$">, we have
- </FONT><!-- MATH
- \begin{displaymath}
-({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="223" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img197.gif"
- ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = b_{k2}\, (\widehat{\mathop{\rm Curl}}\, \hat{v})_{kl}\,b_{l3}
-$">
-</DIV><P></P><FONT SIZE="-1">
- Writing this out and recalling that <!-- MATH
- $\mathop{\rm Curl}v$
- -->
-<IMG
- WIDTH="52" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img183.gif"
- ALT="$ \mathop{\rm Curl}v$"> is skew symmetric, yields
- </FONT><!-- MATH
- \begin{displaymath}
-({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\widehat{\mathop{\rm Curl}} \,\hat{v})_{12}
- -(b_{12}b_{33} - b_{32}b_{13})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{31}
- +(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="739" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img198.gif"
- ALT="$\displaystyle ({\mathop{\rm Curl}v})_{23} = (b_{12}b_{23} - b_{22}b_{13})(\wide...
-...
-+(b_{22}b_{33} - b_{32}b_{23})(\widehat{\mathop{\rm Curl}}\, \hat{v})_{23}\,,
-$">
-</DIV><P></P><FONT SIZE="-1">
- and with (<A HREF="node2.html#eq:_curl2Curl">18</A>) this is equal to the determinant of
- </FONT><!-- MATH
- \begin{displaymath}
-\mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm curl}}\, v)_1 & b_{12} & b_{13} \\
- (\widehat{\mathop{\rm curl}}\, v)_2 & b_{22} & b_{23} \\
- (\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33}
- \end{array}\right) \,.
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="253" HEIGHT="93" ALIGN="MIDDLE" BORDER="0"
- SRC="img199.gif"
- ALT="$\displaystyle \mathrm{M_1} := \left(\begin{array}{ccc} (\widehat{\mathop{\rm cu...
-...
-(\widehat{\mathop{\rm curl}}\, v)_3 & b_{32} & b_{33}
-\end{array}\right) \,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- The proof for the other components follows analogously.
-
-</FONT>
-<P>
-<FONT SIZE="-1">In the next proposition, we state an alternative, equivalent, formula for the transformation of the curl
- (e. g. used by Demkovicz in [<A
- HREF="node4.html#Demko3d">12</A>])
- </FONT><P>
-<DIV><A NAME="prop:_Demko_curl"><B>P<SMALL>ROPOSITION</SMALL> 4</B></A>
-For a vector field <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$"> on the tetrahedron <!-- MATH
- $K=F_K(\hat{K})$
- -->
-<IMG
- WIDTH="100" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img83.gif"
- ALT="$ K = F_K(\hat{K})$">, defined by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) of a reference
- field <!-- MATH
- $\hat{\underline v}$
- -->
-<IMG
- WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img182.gif"
- ALT="$ \hat{\underline v}$"> on <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">, we have
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_3d_curl_trafo_Demko"></A><!-- MATH
- \begin{equation}
-\mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="271" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
- SRC="img200.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{1}{\det B_K}\,B_K\,(\widehat{\mathop{\rm curl}}\,\hat{\underline v} \circ F_K^{-1})\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(19)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- The transformation formula (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) can be proven componentwise, and
- we will only carry out the proof for the first vector component <!-- MATH
- $(\mathop{\rm curl}\underline v)_1$
- -->
-<IMG
- WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img201.gif"
- ALT="$ (\mathop{\rm curl}\underline v)_1$">. The proofs for the other components
- follow analogously.
-<BR>
-The identity (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) reads for the first vector component
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_1.comp_of_curl_trafo"></A><!-- MATH
- \begin{equation}
-(\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="334" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
- SRC="img202.gif"
- ALT="$\displaystyle (\mathop{\rm curl}\underline v)_1 = \frac{1}{\det B_K} (B_K)_{1j} ((\widehat{\mathop{\rm curl}}\,\hat{\underline v})_j \circ F^{-1})\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(20)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- Referring to (<A HREF="node2.html#eq:_curl_trafo_3d">17</A>), we show that the right hand side of (<A HREF="node2.html#eq:_1.comp_of_curl_trafo">20</A>) equals
- <!-- MATH
- $\det \mathrm{M_1}$
- -->
-<IMG
- WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img203.gif"
- ALT="$ \det \mathrm{M_1}$">.
- For this, we expand <!-- MATH
- $\det \mathrm{M_1}$
- -->
-<IMG
- WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img203.gif"
- ALT="$ \det \mathrm{M_1}$"> to
- </FONT><!-- MATH
- \begin{displaymath}
-\det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det\mathcal{B}^{inv}_{11}
- -(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det\mathcal{B}^{inv}_{21}
- +(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="505" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img204.gif"
- ALT="$\displaystyle \det \mathrm{M_1} = (\widehat{\mathop{\rm curl}}\, \hat{\underlin...
-...ehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det\mathcal{B}^{inv}_{31} \,,
-$">
-</DIV><P></P><FONT SIZE="-1">
- where <!-- MATH
- $\mathcal{B}^{inv}_{ij}$
- -->
-<IMG
- WIDTH="39" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img205.gif"
- ALT="$ \mathcal{B}^{inv}_{ij}$"> is the <!-- MATH
- $2 \times 2$
- -->
-<IMG
- WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img206.gif"
- ALT="$ 2 \times 2$">-matrix arising from <IMG
- WIDTH="39" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img207.gif"
- ALT="$ B_K^{-1}$"> when cancelling its i-th row and its j-th column.
-<BR>
-We recall the formula for the inverse of a matrix <!-- MATH
- $A \in \mathbb{R}^{3\times 3}$
- -->
-<IMG
- WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img208.gif"
- ALT="$ A \in \mathbb{R}^{3\times 3}$">
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Ainv"></A><!-- MATH
- \begin{equation}
-(A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="251" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
- SRC="img209.gif"
- ALT="$\displaystyle (A^{-1})_{ij} = \frac{1}{\det A} (-1)^{i+j} \det \mathcal{A}_{ji} \,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(21)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- where <!-- MATH
- $\mathcal{A}_{ij}$
- -->
-<IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img210.gif"
- ALT="$ \mathcal{A}_{ij}$"> is the <!-- MATH
- $2 \times 2$
- -->
-<IMG
- WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img206.gif"
- ALT="$ 2 \times 2$">-matrix arising from <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img211.gif"
- ALT="$ A$"> when cancelling its i-th row and its j-th column.
-<BR>
-Replacing <IMG
- WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img165.gif"
- ALT="$ B_K$"> in the right hand side of (<A HREF="node2.html#eq:_3d_curl_trafo_Demko">19</A>) by the expression (<A HREF="node2.html#eq:_Ainv">21</A>)
- for <!-- MATH
- $A = B_K^{-1}$
- -->
-<IMG
- WIDTH="75" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img212.gif"
- ALT="$ A = B_K^{-1}$">, we get
- </FONT><!-- MATH
- \begin{displaymath}
-\frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \mathcal{B}^{inv}_{j1} (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_j
- = (\widehat{\mathop{\rm curl}}\, \hat{\underline v})_1 \det \mathcal{B}^{inv}_{11}
- -(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_2 \det \mathcal{B}^{inv}_{21}
- +(\widehat{\mathop{\rm curl}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,.
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="834" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
- SRC="img213.gif"
- ALT="$\displaystyle \frac{1}{\det B_K}\,\frac{1}{\det B_K^{-1}} (-1)^{1+j} \det \math...
-...l}}\, \hat{\underline v})_3 \det \mathcal{B}^{inv}_{31} = \det \mathrm{M_1}\,.
-$">
-</DIV><P></P><FONT SIZE="-1">
-
-</FONT>
-<P>
-<FONT SIZE="-1"></FONT>
-<H2><A NAME="SECTION00022000000000000000"></A> <A NAME="sec:_parallelo"></A>
-<BR>
-2.2 Nédélec Elements on affine quadrilateral or hexahedral grids
-</H2><FONT SIZE="-1">
- We want to present the ingredients for Nédélec's finite elements of first type on
- grids consisiting of parallelograms (in 2d) or the respective objects in 3d, so called parallelotops (cf. section
- <I>FE built on cubes</I> in [<A
- HREF="node4.html#Ned1">8</A>]). Such grids consist
- of elements <IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img214.gif"
- ALT="$ C$"> that are affine images of the square or cubic reference element <!-- MATH
- $\hat{C} = [0,1]^d$
- -->
-<IMG
- WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img215.gif"
- ALT="$ \hat{C} = [0,1]^d$">:
- </FONT><!-- MATH
- \begin{displaymath}
-C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="317" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img216.gif"
- ALT="$\displaystyle C = F_C(\hat{C}) \quad C \ni x = B_C \hat{x} + \underline b_C \,, \hat{x} \in \hat{C}\,.
-$">
-</DIV><P></P>
-<P>
-
-<H3><A NAME="SECTION00022100000000000000">
-2.2.1 Polynomial spaces on the reference element</A>
-</H3>
-<P>
-<FONT SIZE="-1">In order to introduce the function spaces needed for the construction of Nédélec's finite elements,
- let us define some spaces of vector-valued polynomials
- </FONT><P>
-<DIV><B>D<SMALL>EFINITION</SMALL> 7</B>
-<!-- MATH
- $\mathcal{Q}_{l,m}$
- -->
-<IMG
- WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img217.gif"
- ALT="$ \mathcal{Q}_{l,m}$"> are the spaces of polynomials on the reference square <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$"> with maximal degree <IMG
- WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img219.gif"
- ALT="$ l$"> in <IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img220.gif"
- ALT="$ \hat{x}_1$"> and
- <IMG
- WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img221.gif"
- ALT="$ m$"> in <IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img222.gif"
- ALT="$ \hat{x}_2$">.
-<BR> <!-- MATH
- $\mathcal{Q}_{l,m,n}$
- -->
-<IMG
- WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img223.gif"
- ALT="$ \mathcal{Q}_{l,m,n}$"> are the spaces of polynomials on the reference cube <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$"> with maximal degree <IMG
- WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img219.gif"
- ALT="$ l$">
- in <IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img220.gif"
- ALT="$ \hat{x}_1$">, <IMG
- WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img221.gif"
- ALT="$ m$"> in <IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img222.gif"
- ALT="$ \hat{x}_2$"> and <IMG
- WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img123.gif"
- ALT="$ n$"> in <IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img224.gif"
- ALT="$ \hat{x}_3$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">The spaces <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img84.gif"
- ALT="$ \hat{R}$"> for the reference shape functions now are in 2d
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_2d_spaces_for_quads"></A><!-- MATH
- \begin{equation}
-\mathcal{P}^k = \left\{ \hat{\underline u} =
- \left(\begin{array}{cc} \hat{u}_1 \\\hat{u}_2 \end{array}\right): \quad
- \hat{u}_1 \in \mathcal{Q}_{k-1,k}\,,
- \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,
-
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="402" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img225.gif"
- ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
-...t{u}_1 \in \mathcal{Q}_{k-1,k}\,, \hat{u}_2 \in \mathcal{Q}_{k,k-1} \right\}\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(22)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-</FONT>
-<P>
-<FONT SIZE="-1">and in 3d
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_3d_spaces_for_quads"></A><!-- MATH
- \begin{equation}
-\mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{ccc}
- \hat{u}_1 \\\hat{u}_2 \\\hat{u}_3 \end{array}\right):\quad
- \hat{u}_1 \in \mathcal{Q}_{k-1,k,k}\,,
- \hat{u}_2 \in \mathcal{Q}_{k,k-1,k}\,,
- \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.
-
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="546" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
- SRC="img226.gif"
- ALT="$\displaystyle \mathcal{P}^k = \left\{ \hat{\underline u} = \left(\begin{array}{...
-...}_2 \in \mathcal{Q}_{k,k-1,k}\,, \hat{u}_3 \in \mathcal{Q}_{k,k,k-1}\right\}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(23)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-</FONT>
-<P>
-<FONT SIZE="-1">We renounce an example, since it is quite evident, what these spaces look like for a specific <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$">.
-</FONT>
-<P>
-
-<H3><A NAME="SECTION00022200000000000000"></A>
-<A NAME="subsubsect:_dofs_on_quads"></A>
-<BR>
-2.2.2 Degrees of freedom on the reference element
-</H3><FONT SIZE="-1">
- We start with the degrees of freedoms on the reference square <!-- MATH
- $\hat{C}\subset \mathbb{R}^2$
- -->
-<IMG
- WIDTH="64" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img227.gif"
- ALT="$ \hat{C}\subset \mathbb{R}^2$">:
- </FONT><P>
-<DIV><A NAME="def:_quad-dofs_2d"><B>D<SMALL>EFINITION</SMALL> 8</B></A>
-Let <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$"> denote the reference square and <!-- MATH
- $\hat{\underline t}$
- -->
-<IMG
- WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img125.gif"
- ALT="$ \hat{\underline t}$"> the tangent as defined in convention <A HREF="node1.html#def:_tangent">1</A>.
- The set of degrees of freedom <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> on <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$"> in the 2d case consists of the linear functionals
- <DL>
-<DT><STRONG>edge dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
- \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img229.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
-... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$">. We have a total of <IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img230.gif"
- ALT="$ 4k$"> of edge dofs.
-
-<P>
-</DD>
-<DT><STRONG>inner dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x}\,, \quad
- \forall\, \hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
- \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="566" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
- SRC="img231.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
-... }_1\in\mathcal{Q}_{k-2,k-1}\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-We have a total of <IMG
- WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img232.gif"
- ALT="$ 2k(k-1)$"> of inner dofs.
-
-<P>
-</DD>
-</DL></DIV><P></P>
-
-<P>
-<P>
-<DIV><A NAME="def:_quad-dofs_3d"><B>D<SMALL>EFINITION</SMALL> 9</B></A>
-Let <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$"> denote the reference cube, <!-- MATH
- $\hat{\underline t}$
- -->
-<IMG
- WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img125.gif"
- ALT="$ \hat{\underline t}$"> the tangent to an edge as defined in convention <A HREF="node1.html#def:_tangent">1</A>
- and <!-- MATH
- $\hat{\underline n}$
- -->
-<IMG
- WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img131.gif"
- ALT="$ \hat{\underline n}$"> the outward unit normal vector to a face.
- The set of degrees of freedom <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> on <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$"> in the 3d case consists of the linear functionals
- <DL>
-<DT><STRONG>edge dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underline t}\cdot \hat{\underline u})\, \hat{\varphi }\,d\hat{s}\,, \quad
- \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img229.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{e}} (\hat{\underli...
-... }\,d\hat{s}\,, \quad \forall\, \hat{\varphi } \in \mathbb{P}_{k-1}(\hat{e})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$">. We have a total of <IMG
- WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img233.gif"
- ALT="$ 12k$"> of edge dofs.
-
-<P>
-</DD>
-<DT><STRONG>face dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underline u}\wedge \hat{\underline n})\cdot \hat{\underline \varphi }\,d\hat{a} \,,\quad
- \forall \,\hat{\underline \varphi } = \left(\begin{array}{cc} \hat{\varphi }_1 \\\hat{\varphi }_2 \end{array}\right) \,,
- \quad\hat{\varphi }_1\in\mathcal{Q}_{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="651" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
- SRC="img234.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{f}} (\hat{\underli...
-..._{k-2,k-1}(\hat{f})\,,\quad\hat{\varphi }_2\in\mathcal{Q}_{k-1,k-2}(\hat{f})\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for every face <IMG
- WIDTH="17" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img134.gif"
- ALT="$ \hat{f}$"> of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$">. We have a total of <!-- MATH
- $6\cdot 2k(k-1)$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img235.gif"
- ALT="$ 6\cdot 2k(k-1)$"> of face dofs.
-
-<P>
-</DD>
-<DT><STRONG>inner dofs</STRONG></DT>
-<DD><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underline u}\cdot \hat{\underline \varphi }\,d\hat{x} \,,\quad
- \forall\, \hat{\underline \varphi } = \left(\begin{array}{ccc} \hat{\varphi }_1 \\\hat{\varphi }_2 \\
- \hat{\varphi }_3\end{array}\right) \,,\quad\hat{\varphi }_1\in\mathcal{Q}_{k-1,k-2,k-2}\,,\quad\hat{\varphi _2}\in\mathcal{Q}_{k-2,k-1,k-2}\,,
- \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="793" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
- SRC="img236.gif"
- ALT="$\displaystyle \hat{\alpha}(\hat{\underline u}) := \int_{\hat{C}} \hat{\underlin...
-...athcal{Q}_{k-2,k-1,k-2}\,, \quad\hat{\varphi _3}\in\mathcal{Q}_{k-2,k-2,k-1}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-We have a total of <IMG
- WIDTH="86" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img237.gif"
- ALT="$ 3k(k-1)^2$"> of inner dofs.
-
-</DD>
-</DL></DIV><P></P>
-
-<P>
-<P>
-<DIV><B>E<SMALL>XAMPLE</SMALL> 4</B>
-Proceeding the same way as in example <A HREF="node2.html#ex:_shape_functions">3</A> for a triangular reference element, we obtain
-the reference shape functions of lowest
-order on the square <IMG
- WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img238.gif"
- ALT="$ [0,1]^2$">. For the unit tangents as in convention <A HREF="node1.html#def:_tangent">1</A>
-<!-- MATH
- \begin{displaymath}
-\hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\0 \end{array}\right)\,,\quad
- \hat{\underline t}_1 = \left(\begin{array}{cc} 0 \\1 \end{array}\right)\,,\quad
- \hat{\underline t}_2 = \left(\begin{array}{cc} -1 \\0 \end{array}\right)\,, \quad
- \hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\-1 \end{array}\right)\,,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="487" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img239.gif"
- ALT="$\displaystyle \hat{\underline t}_0 = \left(\begin{array}{cc} 1 \\ 0 \end{array...
-...
-\hat{\underline t}_3 = \left(\begin{array}{cc} 0 \\ -1 \end{array}\right)\,,
-$">
-</DIV><P></P>
-they read
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation}
-\hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\0 \end{array}\right)\,,\quad
- \hat{\underline N}_1 = \left(\begin{array}{cc} 0 \\\hat{x} \end{array}\right)\,,\quad
- \hat{\underline N}_2 = \left(\begin{array}{cc} -\hat{y} \\0 \end{array}\right)\,,\quad
- \hat{\underline N}_3 = \left(\begin{array}{cc} 0 \\\hat{x}-1 \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="573" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
- SRC="img240.gif"
- ALT="$\displaystyle \hat{\underline N}_0 = \left(\begin{array}{cc} 1-\hat{y} \\ 0 \e...
-...\underline N}_3 = \left(\begin{array}{cc} 0 \\ \hat{x}-1 \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(24)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-
-<H3><A NAME="SECTION00022300000000000000">
-2.2.3 Transformation of the vector field</A>
-</H3><FONT SIZE="-1">
-Since the elements of the considered grids are still affine images of the reference element, we can use the Piola transformation
-(<A HREF="node2.html#eq:_Piola_tri">10</A>) to transform vector fields and the results stated in sections
-<A HREF="node2.html#sec:_Piola_for_triangles">2.1.3</A> - <A HREF="node2.html#subsubsect:_3d-curl_on_tria">2.1.5</A> can be carried over one to one.
-</FONT>
-<H2><A NAME="SECTION00023000000000000000">
-2.3 Construction of Nédélec elements on bi- or trilinear elements</A>
-</H2><FONT SIZE="-1">
- We now want to consider grids that are composed of elements that are a bi- resp. trilinear images <!-- MATH
- $F_C(\hat{C})$
- -->
-<IMG
- WIDTH="56" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img241.gif"
- ALT="$ F_C(\hat{C})$"> of the reference
- element <!-- MATH
- $\hat{C} = [0,1]^d$
- -->
-<IMG
- WIDTH="86" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img215.gif"
- ALT="$ \hat{C} = [0,1]^d$">.
- The main difference here is, that the jacobian <!-- MATH
- $\hat{D}F_C(\hat{x})$
- -->
-<IMG
- WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img242.gif"
- ALT="$ \hat{D}F_C(\hat{x})$"> of the element map <IMG
- WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img243.gif"
- ALT="$ F_C$"> is not
- constant, and we have to use Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) to
- transform vector fields.
-<BR></FONT>
-<P>
-
-<H3><A NAME="SECTION00023100000000000000">
-2.3.1 Bilinear elements in 2d</A>
-</H3><FONT SIZE="-1">
- The polynomial spaces <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine quadrilateral elements.
-<BR>
-A transformed vector field on a general element is now defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>)
- </FONT><!-- MATH
- \begin{displaymath}
-\underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="212" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img244.gif"
- ALT="$\displaystyle \underline v(x) = (\hat{D}F_C^{-T} \hat{\underline v}_i) \circ F_C^{-1} (x)$">
-</DIV><P></P><FONT SIZE="-1"> of a vector field on the reference element. Note that
- the jacobian <!-- MATH
- $\hat{D}F_C(\hat{x})$
- -->
-<IMG
- WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img242.gif"
- ALT="$ \hat{D}F_C(\hat{x})$"> is not constant in this case. In contrast to the case of affine elements, the gradient <!-- MATH
- $D\underline v$
- -->
-<IMG
- WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img174.gif"
- ALT="$ D\underline v$"> does not transform according to
- formula (<A HREF="node2.html#eq:_gradient_trafo_2d">12</A>). Non-vanishing second derivatives of <!-- MATH
- $\hat{D}F_C(\hat{x})$
- -->
-<IMG
- WIDTH="67" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img242.gif"
- ALT="$ \hat{D}F_C(\hat{x})$"> appear in the transformation rule
- for gradients of vector fields.
- This requires a new approach to express <!-- MATH
- $\mathop{\rm curl}\underline v$
- -->
-<IMG
- WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img245.gif"
- ALT="$ \mathop{\rm curl}\underline v$"> in terms of
- <!-- MATH
- $\widehat{\mathop{\rm curl}}\,\hat{\underline v}$
- -->
-<IMG
- WIDTH="47" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img246.gif"
- ALT="$ \widehat{\mathop{\rm curl}}\,\hat{\underline v}$">.
- Nevertheless, it can be shown that the curl of a vector field transforms analogously to the case of affine elements.
-</FONT>
-<P>
-<P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 5</B>
-Let <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$"> be the reference element <IMG
- WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img238.gif"
- ALT="$ [0,1]^2$"> and <IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img214.gif"
- ALT="$ C$"> a bilinear image of <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$">. If the vector field <!-- MATH
- $\underline v(x)$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img187.gif"
- ALT="$ \underline v(x)$">
- transforms according to the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>), then the transformation of the curl obeys
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="370" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img247.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v(x) = (\det \hat{D}F)^{-1} \widehat{\mathop{\rm curl}}\, \hat{\underline v}(\hat{x})\,, \qquad x = F(\hat{x})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-as in the affine case.
-<BR> </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- In this proof, the mapped element <IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img214.gif"
- ALT="$ C$"> will be fixed, so for simplicity we write <IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img248.gif"
- ALT="$ F$"> for <IMG
- WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img243.gif"
- ALT="$ F_C$">.
-<BR>
-First note that <!-- MATH
- $(\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$
- -->
-<IMG
- WIDTH="242" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img249.gif"
- ALT="$ (\hat{D}F(F^{-1}(x)))^{-1} = D(F^{-1})(x)$">. We use the notation <!-- MATH
- $D(F^{-1})_{ij}(x)=
-\frac{\partial \hat{x}_i}{\partial x_j}(x)$
- -->
-<IMG
- WIDTH="174" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img250.gif"
- ALT="$ D(F^{-1})_{ij}(x)=
-\frac{\partial \hat{x}_i}{\partial x_j}(x)$"> and imlicit summation to rewrite the Piola transformation of the vector field componentwise
- </FONT><!-- MATH
- \begin{displaymath}
-v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="317" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
- SRC="img251.gif"
- ALT="$\displaystyle v_i(x) = \frac{\partial \hat{x}_j}{\partial x_i}(x) \,\hat{\underline v}_j (F^{-1}(x)) \,, \qquad i=1,2\,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- In the case of affine elements, i. e. for constant jacobian, we have
-</FONT>
-<P>
-<P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- \frac{\partial v_2}{\partial x_1} &= \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
- \frac{\partial v_1}{\partial x_2} &= \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,,
- \end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="398" HEIGHT="90" BORDER="0"
- SRC="img252.gif"
- ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part...
-... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,, \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-<FONT SIZE="-1">whereas for non-constant jacobian we have
-</FONT>
-<P>
-<P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- \frac{\partial v_2}{\partial x_1} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x)\, \hat{\underline v}_i(F^{-1}(x)) +
- \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x)) \\
- \frac{\partial v_1}{\partial x_2} &= \frac{\partial ^2 \hat{x}_i}{\partial x_1\partial x_2}(x) \,\hat{\underline v}_i(F^{-1}(x)) +
- \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,.
- \end{split}
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="493" HEIGHT="95" BORDER="0"
- SRC="img253.gif"
- ALT="\begin{displaymath}\begin{split}\frac{\partial v_2}{\partial x_1} &= \frac{\part...
-... }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<P>
-<FONT SIZE="-1">We see that in <I>both</I> cases we have
- </FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1} - \frac{\partial v_1}{\partial x_2} =
- \frac{\partial \hat{x}_i}{\partial x_2}(x) \,\frac{\partial }{\partial x_1}\hat{\underline v}_i(F^{-1}(x))
- - \frac{\partial \hat{x}_i}{\partial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="548" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
- SRC="img254.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \frac{\partial v_2}{\partial x_1}...
-...ial x_1}(x) \,\frac{\partial }{\partial x_2}\hat{\underline v}_i(F^{-1}(x)) \,
-$">
-</DIV><P></P><FONT SIZE="-1">
- that is, the second derivatives cancel out in the expression for the curl and the curl in the non-affine case transforms equally
- to the curl in the affine case.
-
-</FONT>
-<P>
-
-<H3><A NAME="SECTION00023200000000000000">
-2.3.2 Trilinear elements in 3d</A>
-</H3><FONT SIZE="-1">
- The polynomial spaces <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$"> and the dofs remain the same as in the case of affine hexahedral elements.
-<BR>
-The
- vector field on a genereal element is defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>).
-<BR>
-The problem of the non-vanishing second derivatives of the jacobian <!-- MATH
- $D(F_C^{-1})(x)$
- -->
-<IMG
- WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img255.gif"
- ALT="$ D(F_C^{-1})(x)$"> arises again, and we cannot generalize the results
- from the affine case straight away.
-<BR>
-But analogously to the 2d case, one can check that in in the transformation rule for expressions
- <!-- MATH
- $\frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$
- -->
-<IMG
- WIDTH="78" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img256.gif"
- ALT="$ \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}$">, <!-- MATH
- $i,j = 1,2,3$
- -->
-<IMG
- WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img257.gif"
- ALT="$ i,j = 1,2,3$">,
- which define the curl-operator, the terms containing second derivatives vanish. We have therefore again the transformation rule
- (<A HREF="node2.html#eq:_Curl_trafo">16</A>) for the skew matrix <!-- MATH
- $\mathop{\rm Curl}v = Dv^T - Dv$
- -->
-<IMG
- WIDTH="155" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img258.gif"
- ALT="$ \mathop{\rm Curl}v = Dv^T - Dv$">:
- </FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm Curl}}\,\hat{v} \,\hat{D}F_C^{-1}) \circ F_C^{-1})(x)
- = (DF_C^{-1})^T(x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="673" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img259.gif"
- ALT="$\displaystyle \mathop{\rm Curl}v (x) = ((\hat{D}F_C^{-T} \widehat{\mathop{\rm C...
-...x)\, (\widehat{\mathop{\rm Curl}}\,\hat{v}\circ F_C^{-1})(x)\,DF_C^{-1}(x) \,.
-$">
-</DIV><P></P>
-<P>
-<FONT SIZE="-1">It follows that the following proposition can be proved analogously to the case of affine elements (replace there <IMG
- WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img260.gif"
- ALT="$ B_C$"> by
- <!-- MATH
- $\hat{D}F_C(\hat(x))$
- -->
-<IMG
- WIDTH="81" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
- SRC="img261.gif"
- ALT="$ \hat{D}F_C(\hat(x))$">).
- </FONT><P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 6</B>
-Let the vector field <!-- MATH
- $\underline v(x)$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img187.gif"
- ALT="$ \underline v(x)$"> on a trilinear image <!-- MATH
- $C = F_C(\hat{C})$
- -->
-<IMG
- WIDTH="93" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img262.gif"
- ALT="$ C = F_C(\hat{C})$"> be defined by the Piola transformation of a
- reference field <!-- MATH
- $\hat{\underline v}(\hat{x})$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img188.gif"
- ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
- WIDTH="20" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img218.gif"
- ALT="$ \hat{C}$">. The transformation formula for the curl then reads
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\mathop{\rm curl}\underline v =
- \left(\frac{1}{\det \hat{D}F_C}\,\hat{D}F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.
-
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="305" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
- SRC="img263.gif"
- ALT="$\displaystyle \mathop{\rm curl}\underline v = \left(\frac{1}{\det \hat{D}F_C}\,...
-...F_C\, \widehat{\mathop{\rm curl}}\,\hat{\underline v} \right) \circ F_C^{-1}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1"></FONT>
-<H2><A NAME="SECTION00024000000000000000">
-2.4 Construction of global shape functions</A>
-</H2><FONT SIZE="-1">
- In the previous sections we have introduced function spaces and degrees of freedom, which, together with the Piola transformation,
- will allow us to define an <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element method.
- Indeed, in [<A
- HREF="node4.html#Ned1">8</A>], Nédélec shows the invariance of the spaces
- <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> and <!-- MATH
- $\mathcal{Q}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img264.gif"
- ALT="$ \mathcal{Q}^k$"> under Piola transformation of the vector field, as well as the unisolvence of the set of
- degrees of freedom <!-- MATH
- $\mathcal{A}$
- -->
-<IMG
- WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$ \mathcal{A}$"> from sections <A HREF="node2.html#subsubsect:_dofs_on_tria">2.1.2</A> and <A HREF="node2.html#subsubsect:_dofs_on_quads">2.2.2</A>
- (for details, see [<A
- HREF="node4.html#Ned1">8</A>], Section 1.2, Theorem 1 and Section 2, Theorem 5).
- This leads to the fact that <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions can be defined by mapping elementwise the reference
- shape functions with the Piola transformation <!-- MATH
- $\mathcal{P}_K$
- -->
-<IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img265.gif"
- ALT="$ \mathcal{P}_K$">.
- However, we must pay some care to the orientation of an interface on which the moments defining the degrees of freedom are based.
- For the 2d case, we will illustrate in this section how we must take into account the <I>orientation of an edge</I>
- in the definition of the respective element edge shape function, in order to get an <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming finite element space
- of global shape functions.
-
-<BR></FONT>
-<P>
-<FONT SIZE="-1">Let <!-- MATH
- $K=F(\hat{K})$
- -->
-<IMG
- WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img266.gif"
- ALT="$ K=F(\hat{K})$"> be an affine or bilinear image of a reference element, <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> one of its edges and <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> the
- corresponding edge on the reference element.
-<BR>
-Let further <!-- MATH
- $[0,|e|] \ni s \mapsto \underline x(s) \in e$
- -->
-<IMG
- WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img267.gif"
- ALT="$ [0,\vert e\vert] \ni s \mapsto \underline x(s) \in e$"> and <!-- MATH
- $[0,|\hat{e}|] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$
- -->
-<IMG
- WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img268.gif"
- ALT="$ [0,\vert\hat{e}\vert] \ni \hat{s} \mapsto \hat{\underline x}(\hat{s}) \in \hat{e}$"> be parametrizations
- with respect to the arc length of <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> and <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> respectively. We can assume that these parametrizations endow the edges with a
- counterclockwise orientation. Then, the unit tangent vectors <!-- MATH
- $\underline t$
- -->
-<IMG
- WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img12.gif"
- ALT="$ \underline t$"> and <!-- MATH
- $\hat{\underline t}$
- -->
-<IMG
- WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img125.gif"
- ALT="$ \hat{\underline t}$"> are given by
- <!-- MATH
- $\frac{d \underline x}{ds}$
- -->
-<IMG
- WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
- SRC="img269.gif"
- ALT="$ \frac{d \underline x}{ds}$"> and <!-- MATH
- $\frac{d \hat{\underline x}}{d\hat{s}}$
- -->
-<IMG
- WIDTH="25" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
- SRC="img270.gif"
- ALT="$ \frac{d \hat{\underline x}}{d\hat{s}}$">.
-</FONT>
-<P>
-<P>
-<DIV><A NAME="lemma:__v_dot_t_-Trafo"><B>L<SMALL>EMMA</SMALL> 1</B></A>
-Let <!-- MATH
- $\hat{\underline v}(\hat{x})$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img188.gif"
- ALT="$ \hat{\underline v}(\hat{x})$"> be a vector field on the reference element and <!-- MATH
- $\underline v(x)$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img187.gif"
- ALT="$ \underline v(x)$"> be the corresponding vector field on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$">, defined by the
- Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>). It then holds
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation}
-\underline v\cdot \underline t = \frac{|\hat{e}|}{|e|} (\hat{\underline v}\cdot \hat{\underline t})\,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
- SRC="img271.gif"
- ALT="$\displaystyle \underline v\cdot \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert} (\hat{\underline v}\cdot \hat{\underline t})\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(25)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-where <IMG
- WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img272.gif"
- ALT="$ \vert\hat{e}\vert$"> and <IMG
- WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img273.gif"
- ALT="$ \vert e\vert$"> denote the length of the edges <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> and <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- With
- </FONT><!-- MATH
- \begin{displaymath}
-(\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="300" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
- SRC="img274.gif"
- ALT="$\displaystyle (\underline v(x))_i = (D(F^{-1})^T \hat{\underline v})_i = \frac{\partial \hat{x}_j}{\partial x_i}(x) \hat{\underline v}_j(\hat{x})\,
-$">
-</DIV><P></P><FONT SIZE="-1">
- and <!-- MATH
- $\hat{x}_j = \hat{x}_j(\underline x(s))$
- -->
-<IMG
- WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img275.gif"
- ALT="$ \hat{x}_j = \hat{x}_j(\underline x(s))$"> and <!-- MATH
- $\hat{x}_j = \hat{x}_j(\hat{s}(s))$
- -->
-<IMG
- WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img276.gif"
- ALT="$ \hat{x}_j = \hat{x}_j(\hat{s}(s))$"> on the edges, we have
- </FONT><!-- MATH
- \begin{displaymath}
-\underline v\cdot \underline t= \underline v \cdot \frac{d \underline x}{ds} = \left( \hat{\underline v}_j\frac{\partial \hat{x}_j}{\partial x_i}\right) (x)\frac{dx_i}{ds}
- = \hat{\underline v}_j \frac{d \hat{x}_j}{ds} = \hat{\underline v}_j \frac{d \hat{x}_j}{d\hat{s}} \frac{d\hat{s}}{ds}
- = (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds}
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="498" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
- SRC="img277.gif"
- ALT="$\displaystyle \underline v\cdot \underline t= \underline v \cdot \frac{d \under...
-...t{s}}{ds}
-= (\hat{\underline v}\cdot \hat{\underline t}) \frac{d\hat{s}}{ds}
-$">
-</DIV><P></P><FONT SIZE="-1">
- and with <!-- MATH
- $\frac{d\hat{s}}{ds}=\frac{|\hat{e}|}{|e|}$
- -->
-<IMG
- WIDTH="64" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img278.gif"
- ALT="$ \frac{d\hat{s}}{ds}=\frac{\vert\hat{e}\vert}{\vert e\vert}$"> the lemma follows.
-
-</FONT>
-<P>
-<FONT SIZE="-1">As a consequence, we have
-</FONT>
-<P>
-<P>
-<DIV><A NAME="prop:_Dof_invariance"><B>P<SMALL>ROPOSITION</SMALL> 7</B></A> (Invariance of the edge dofs)
-Let the vector field <!-- MATH
- $\underline v(x)$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img187.gif"
- ALT="$ \underline v(x)$"> on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> be defined by the Piola transformation (<A HREF="node2.html#eq:_Piola">9</A>) of a reference vector field
- <!-- MATH
- $\hat{\underline v}(\hat{x})$
- -->
-<IMG
- WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img188.gif"
- ALT="$ \hat{\underline v}(\hat{x})$"> on <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$">. Then, the functionals (<I>edge dofs</I>)
- <!-- MATH
- $\alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$
- -->
-<IMG
- WIDTH="178" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img279.gif"
- ALT="$ \alpha^{[K]}(\underline u) := \int_e (\underline v\cdot \underline t)\varphi \,ds$"> are invariant in the sense of
- <!-- MATH
- \begin{displaymath}
-\alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline t)\varphi \,ds =
- \int_{\hat{e}} (\hat{\underline v} \cdot \hat{\underline t}) \hat{\varphi } \, d\hat{s}\, = \hat{\alpha}(\hat{\underline u})\,,
- \qquad \forall\, \hat{\varphi } \in
- \mathbb{P}_{k-1}(\hat{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="704" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
- SRC="img280.gif"
- ALT="$\displaystyle \alpha^{[K]}(\underline u) = \int_e (\underline v\cdot \underline...
-...{e})\,, \quad \varphi = \hat{\varphi } \circ F^{-1} \in \mathbb{P}_{k-1}(e)\,.
-$">
-</DIV><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">Let now <!-- MATH
- $K_- = F_-(\hat{K})$
- -->
-<IMG
- WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img281.gif"
- ALT="$ K_- = F_-(\hat{K})$"> and <!-- MATH
- $K_+ = F_+(\hat{K})$
- -->
-<IMG
- WIDTH="108" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img282.gif"
- ALT="$ K_+ = F_+(\hat{K})$"> be two neighbouring triangles with common edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">. Let <!-- MATH
- $\underline N$
- -->
-<IMG
- WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img283.gif"
- ALT="$ \underline N$"> be the global edge shape
- function that 'lives' on <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">. By <!-- MATH
- $\underline N_-$
- -->
-<IMG
- WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img284.gif"
- ALT="$ \underline N_-$"> and <!-- MATH
- $\underline N_+$
- -->
-<IMG
- WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img285.gif"
- ALT="$ \underline N_+$"> we denote the restriction of <!-- MATH
- $\underline N$
- -->
-<IMG
- WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img283.gif"
- ALT="$ \underline N$"> to <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> and <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$"> respectively.
- Let <!-- MATH
- $e_+ =F_+(\hat{e}_i)$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img286.gif"
- ALT="$ e_+ =F_+(\hat{e}_i)$"> and <!-- MATH
- $e_- =F_-(\hat{e}_j)$
- -->
-<IMG
- WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img287.gif"
- ALT="$ e_- =F_-(\hat{e}_j)$">.
- We write <!-- MATH
- $\underline t_+$
- -->
-<IMG
- WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img288.gif"
- ALT="$ \underline t_+ $"> for the tangential unit vector to <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">, oriented
- counterclockwise with respect to <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> and <!-- MATH
- $\underline t_- = -\underline t_+$
- -->
-<IMG
- WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img289.gif"
- ALT="$ \underline t_- = -\underline t_+$">
- for the respective from <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$">. For line integrals over the edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> we write <!-- MATH
- $\int_{e_+}$
- -->
-<IMG
- WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
- SRC="img290.gif"
- ALT="$ \int_{e_+}$"> if we chose the orientation induced by
- <!-- MATH
- $\underline t_+$
- -->
-<IMG
- WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img288.gif"
- ALT="$ \underline t_+ $"> and <!-- MATH
- $\int_{e_-}$
- -->
-<IMG
- WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
- SRC="img291.gif"
- ALT="$ \int_{e_-}$"> for the orientation of <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> induced by <!-- MATH
- $\underline t_-$
- -->
-<IMG
- WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img292.gif"
- ALT="$ \underline t_-$">.
-<BR>
-In order to obtain an <!-- MATH
- $H(\mathop{\rm curl}; \Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming method, proposition <A HREF="node1.html#prop:_no_jumps">1</A> tells us that we must ensure the continuity of the
- tangential components of the global shape functions, that is
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_conformity_of_shape_func"></A><!-- MATH
- \begin{equation}
-\underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="182" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img293.gif"
- ALT="$\displaystyle \underline N_+\cdot\underline t_+ + \underline N_-\cdot\underline t_- = 0\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(26)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- The following lemma will justify the choice of the moments describing the edge dofs. A consequence of the lemma will be,
- that the matching of the local edge dofs <!-- MATH
- $\alpha^{[K_+]}$
- -->
-<IMG
- WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
- SRC="img294.gif"
- ALT="$ \alpha^{[K_+]}$"> and <!-- MATH
- $\alpha^{[K_-]}$
- -->
-<IMG
- WIDTH="47" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
- SRC="img295.gif"
- ALT="$ \alpha^{[K_-]}$"> guarantees
- the pointwise condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
- </FONT><P>
-<DIV><A NAME="lemma:_trace"><B>L<SMALL>EMMA</SMALL> 2</B></A>
-Let <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> denote the reference triangle and <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> one of its edges, parametrized by <!-- MATH
- $\hat{e} \ni \hat{x}(s) := \underline a +
-s\, \hat{\underline t}$
- -->
-<IMG
- WIDTH="143" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img296.gif"
- ALT="$ \hat{e} \ni \hat{x}(s) := \underline a +
-s\, \hat{\underline t}$">.
- Let <!-- MATH
- $\hat{\underline p} \in \mathcal{S}^k$
- -->
-<IMG
- WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img297.gif"
- ALT="$ \hat{\underline p} \in \mathcal{S}^k$">, <!-- MATH
- $\mathcal{S}^k$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img107.gif"
- ALT="$ \mathcal{S}^k$"> as defined in (<A HREF="node2.html#def:_space_Sk">4</A>). It then holds
- <!-- MATH
- \begin{displaymath}
-(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="146" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img298.gif"
- ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})\,.
-$">
-</DIV><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- </FONT><!-- MATH
- \begin{displaymath}
-\hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad \textrm{for } i=1,2,3:
- \quad \hat{p}_i(\hat{x}) = \prod_{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
- \textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="545" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
- SRC="img299.gif"
- ALT="$\displaystyle \hat{\underline p} \in \mathcal{S}^k \quad \Longrightarrow \quad ...
-...{j=1}^3 \hat{x}_j^{k_{ij}}\,, \quad
-\textrm{where } \sum_{j=1}^3 k_{ij} = k\,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- Hence, with the parametrization of <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img127.gif"
- ALT="$ \hat{e}$"> by <!-- MATH
- $\hat{x}(s)$
- -->
-<IMG
- WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img300.gif"
- ALT="$ \hat{x}(s)$">
- </FONT><!-- MATH
- \begin{displaymath}
-\hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
- \hat{\varphi }_{k-1}(s)\,,
-
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="390" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
- SRC="img301.gif"
- ALT="$\displaystyle \hat{p}_i(\hat{x}(s)) = \prod_{j=1}^3 (a_j + s\,\hat{t}_j)^{k_{ij}} = s^k\,\prod_{j=1}^3 \hat{t}_j^{k_{ij}} +
-\hat{\varphi }_{k-1}(s)\,,
-$">
-</DIV><P></P><FONT SIZE="-1">
- with <!-- MATH
- $\hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$
- -->
-<IMG
- WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img302.gif"
- ALT="$ \hat{\varphi }_{k-1}(s) \in \mathbb{P}_{k-1}(\hat{e})$">,
- and
- </FONT><!-- MATH
- \begin{displaymath}
-(\hat{\underline p}\cdot \hat{\underline t})|_{\hat{e}} = s^k\,\sum_{i=1}^3\hat{t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
- \hat{\varphi }_{k-1}(s)\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="317" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
- SRC="img303.gif"
- ALT="$\displaystyle (\hat{\underline p}\cdot \hat{\underline t})\vert _{\hat{e}} = s^...
-...t}_i\left(\prod_{j=1}^3 \hat{t}_j^{k_{ij}}\right) +
-\hat{\varphi }_{k-1}(s)\,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- We observe that the coefficient of <IMG
- WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img304.gif"
- ALT="$ s^k$"> is exactly <!-- MATH
- $\hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$
- -->
-<IMG
- WIDTH="54" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img305.gif"
- ALT="$ \hat{\underline p}(\hat{\underline t}) \cdot \hat{\underline t}$">. By the definition of the space
- <!-- MATH
- $\mathcal{S}^k$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img107.gif"
- ALT="$ \mathcal{S}^k$">, this expression must vanish.
-
-</FONT>
-<P>
-<P>
-<DIV><B>R<SMALL>EMARK</SMALL> 5</B>
-In the case of <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> being a quadrilateral, we have <!-- MATH
- $\hat{R} = \mathcal{P}^k$
- -->
-<IMG
- WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img306.gif"
- ALT="$ \hat{R} = \mathcal{P}^k$">. By the definition of <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$"> we see
- immediately that here also <!-- MATH
- $(\hat{\underline v}\cdot \hat{\underline t})|_{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$
- -->
-<IMG
- WIDTH="139" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img307.gif"
- ALT="$ (\hat{\underline v}\cdot \hat{\underline t})\vert _{\hat{e}} \in \mathbb{P}_{k-1}(\hat{e})$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">The next proposition tells us how exactly to define element shape functions on a mapped element <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> in order to get
- <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming global shape functions.
- </FONT><P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 8</B>
-Condition (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>) is satiesfied, if we define the element shape functions <!-- MATH
- $\underline N_+$
- -->
-<IMG
- WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img285.gif"
- ALT="$ \underline N_+$"> and <!-- MATH
- $\underline N_-$
- -->
-<IMG
- WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img284.gif"
- ALT="$ \underline N_-$">
- by the Piola transformation (<A HREF="node2.html#eq:_Piola_tri">10</A>) and take into account the orientation of the edge <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$">:
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_trafo_shape_functions"></A><!-- MATH
- \begin{equation}
-\underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_+^{-T} \hat{\underline N}_i \,,
- \qquad \underline N_- := -\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.
-
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="508" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img308.gif"
- ALT="$\displaystyle \underline N_+ := \mathcal{P}_+(\hat{\underline N}_i) = \hat{D}F_...
-...\,\mathcal{P}_-(\hat{\underline N}_j) = -\hat{D}F_-^{-T}\hat{\underline N}_j\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(27)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P></DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- Let <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> be the reference element and <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> its affine or bilinear image.
- Let <!-- MATH
- $\underline v := \mathcal{P}_K(\hat{\underline v})$
- -->
-<IMG
- WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img309.gif"
- ALT="$ \underline v := \mathcal{P}_K(\hat{\underline v})$"> be a vector field on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$">, defined by the Piola transformation of a reference vector field
- <!-- MATH
- $\hat{\underline v} \in \hat{R}$
- -->
-<IMG
- WIDTH="50" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img310.gif"
- ALT="$ \hat{\underline v} \in \hat{R}$">. Let <IMG
- WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img66.gif"
- ALT="$ e$"> be one of the edges of <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> and <!-- MATH
- $\underline t$
- -->
-<IMG
- WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img12.gif"
- ALT="$ \underline t$">
- the tangent according to convention <A HREF="node1.html#def:_tangent">1</A>.
-<BR>
-In the case of <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> being a triangle, we have <!-- MATH
- $\hat{R} = \mathcal{R}^k$
- -->
-<IMG
- WIDTH="66" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img91.gif"
- ALT="$ \hat{R} = \mathcal{R}^k$">.
- By the definition of the space <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$">, lemma <A HREF="node2.html#lemma:_trace">2</A> and <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> we can
- conclude that <!-- MATH
- $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
- -->
-<IMG
- WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img311.gif"
- ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
-<BR>
-If <IMG
- WIDTH="23" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
- SRC="img81.gif"
- ALT="$ \hat{K}$"> is a quadrilateral, the previous remark and
- <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A> also tell us that <!-- MATH
- $(\underline v\cdot \underline t)|_{e} \in \mathbb{P}_{k-1}(e)$
- -->
-<IMG
- WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img311.gif"
- ALT="$ (\underline v\cdot \underline t)\vert _{e} \in \mathbb{P}_{k-1}(e)$">.
-<BR>
-Hence the condition
- </FONT><!-- MATH
- \begin{displaymath}
-\int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underline N_-\cdot \underline t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="401" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img312.gif"
- ALT="$\displaystyle \int_{e_+} \left((\underline N_+\cdot \underline t_+) + (\underli...
-...ne t_-)\right)\,\varphi \,ds\,, \qquad \forall\,\varphi \in\mathbb{P}_{k-1}(e)
-$">
-</DIV><P></P><FONT SIZE="-1">
- on the edge moments
- is sufficient for the global edge shape functions to satiesfy (<A HREF="node2.html#eq:_conformity_of_shape_func">26</A>).
- Note that
-<BR><!-- MATH
- $\int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$
- -->
-<IMG
- WIDTH="307" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
- SRC="img313.gif"
- ALT="$ \int_{e_+}(\underline N_-\cdot \underline t_-)\,\varphi \,ds = -\int_{e_-}(\underline N_-\cdot \underline t_-)\,\varphi \,ds$">.
- So, by the definition (<A HREF="node2.html#eq:_trafo_shape_functions">27</A>) of the element shape functions on <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img57.gif"
- ALT="$ K_+$"> resp. on <IMG
- WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img56.gif"
- ALT="$ K_-$">,
- by the invariance of the dofs (proposition <A HREF="node2.html#prop:_Dof_invariance">7</A>) and by the definition of the
- reference shape functions (example <A HREF="node2.html#ex:_shape_functions">3</A>) we have
- </FONT><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \int_{\hat{e}_i} (\hat{\underline N}_i\cdot \hat{\underline t}_i)\hat{\varphi }\,d\hat{s} = 1
- \qquad \textrm{and} \qquad
- \int_{e_-} (\underline N_- \cdot \underline t_-)\varphi \,ds = -\int_{\hat{e}_j} (\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="740" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
- SRC="img314.gif"
- ALT="$\displaystyle \int_{e_+} (\underline N_+ \cdot \underline t_+)\varphi \,ds = \i...
-...\hat{\underline N}_j\cdot \hat{\underline t}_j)\hat{\varphi }\,d\hat{s} = -1\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-
-</FONT>
-<P>
-<FONT SIZE="-1">To close this section, let us make a note on the interpretation of the dofs on an element <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> in the case of lowest order polynomial
- degree. In this case, all dofs are edge dofs, the degrees of freedom are <!-- MATH
- $\hat{\alpha}_j(\hat{\underline
-v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$
- -->
-<IMG
- WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img315.gif"
- ALT="$ \hat{\alpha}_j(\hat{\underline
-v}) = \int_{\hat{e}_j} \hat{\underline v} \cdot \hat{\underline t}_j\,d\hat{s}$"> and the tangential traces of shape functions are constant on each edge.
- Since we require <!-- MATH
- $\hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$
- -->
-<IMG
- WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img316.gif"
- ALT="$ \hat{\alpha}_j(\hat{\underline N}_i) = \delta_{ij}$"> for the reference shape functions, we have
- </FONT><!-- MATH
- \begin{displaymath}
-v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j \cdot \hat{\underline t})\,|\hat{e}_j| = ({\underline N}_j \cdot {\underline t}_j)\, |e_j|\,,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="313" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img317.gif"
- ALT="$\displaystyle v_j = \hat{\alpha}_j(\hat{\underline v}) = (\hat{\underline N}_j ...
-...\hat{e}_j\vert = ({\underline N}_j \cdot {\underline t}_j)\, \vert e_j\vert\,,
-$">
-</DIV><P></P><FONT SIZE="-1">
- where for the last equality we have used lemma <A HREF="node2.html#lemma:__v_dot_t_-Trafo">1</A>. We see that
- the dof <!-- MATH
- $\alpha_j(\underline v)$
- -->
-<IMG
- WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img318.gif"
- ALT="$ \alpha_j(\underline v)$"> 'sitting' on the edge <IMG
- WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img319.gif"
- ALT="$ e_j$"> is the value of the <I>scaled</I> tangential component
- <!-- MATH
- $|e_j|\left(\underline v\cdot \underline t_j\right)|_e$
- -->
-<IMG
- WIDTH="100" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img320.gif"
- ALT="$ \vert e_j\vert\left(\underline v\cdot \underline t_j\right)\vert _e$">.
-<BR> </FONT><P>
-<DIV><B>R<SMALL>EMARK</SMALL> 6</B>
-For the invariance of the edge dofs it is essential that the moments <!-- MATH
- $\alpha^{[K]}$
- -->
-<IMG
- WIDTH="38" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
- SRC="img321.gif"
- ALT="$ \alpha^{[K]}$"> on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$"> are defined by using the
- <I>unit</I> tangent
- vector <!-- MATH
- $\underline t = \frac{|\hat{e}|}{|e|}\,(\hat{D}F)\,\hat{\underline t}$
- -->
-<IMG
- WIDTH="108" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
- SRC="img322.gif"
- ALT="$ \underline t = \frac{\vert\hat{e}\vert}{\vert e\vert}\,(\hat{D}F)\,\hat{\underline t}$"> on <IMG
- WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img86.gif"
- ALT="$ K$">. If not, e. g. if we
- just used the tangent <!-- MATH
- $\tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$
- -->
-<IMG
- WIDTH="87" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img323.gif"
- ALT="$ \tilde{\underline t} = (\hat{D}F)\,\hat{\underline t}$">, we would lose the invariance of the dofs. In that case the dofs would
- scale by a factor depending on the size of the edge or face ([<A
- HREF="node4.html#Ned1">8</A>], remark on p. 326).
-<BR> </DIV><P></P>
-
-<P>
-<FONT SIZE="-1"></FONT>
-<H2><A NAME="SECTION00025000000000000000">
-2.5 Approximation and convergence results</A>
-</H2><FONT SIZE="-1">
- Without going into details, we will cite here some results on approximation properties and convergence of Nédélec FEM of first type.
-
-<BR>
-We are in the setting of a
- <I>conforming</I> FEM and have quasi-optimal approximation properties of the FE-spaces <!-- MATH
- $V_h \subset H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img324.gif"
- ALT="$ V_h \subset H(\mathop{\rm curl};\Omega )$">
- </FONT><!-- MATH
- \begin{displaymath}
-\| \underline u - \Pi_h^k \underline u\|_{H(\mathop{\rm curl}; \Omega )} = C\,\inf_{w\in V_h}\| \underline u - \underline w\|_{H(\mathop{\rm curl}; \Omega )}\,,
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="355" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img325.gif"
- ALT="$\displaystyle \Vert \underline u - \Pi_h^k \underline u\Vert _{H(\mathop{\rm cu...
-...V_h}\Vert \underline u - \underline w\Vert _{H(\mathop{\rm curl}; \Omega )}\,,
-$">
-</DIV><P></P><FONT SIZE="-1">
- where <!-- MATH
- $\Pi_h^k \underline u \in \mathcal{R}^k$
- -->
-<IMG
- WIDTH="82" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img326.gif"
- ALT="$ \Pi_h^k \underline u \in \mathcal{R}^k$"> or <!-- MATH
- $\Pi_h^k \underline u \in \mathcal{P}^k$
- -->
-<IMG
- WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img327.gif"
- ALT="$ \Pi_h^k \underline u \in \mathcal{P}^k$"> respectively,
- denotes the interpolate of <!-- MATH
- $\underline u$
- -->
-<IMG
- WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img35.gif"
- ALT="$ \underline u$"> with regard to the Nédélec dofs: <!-- MATH
- $\alpha(\underline u) = \alpha(\Pi_h^k \underline u)$
- -->
-<IMG
- WIDTH="121" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img328.gif"
- ALT="$ \alpha(\underline u) = \alpha(\Pi_h^k \underline u)$"> for all dofs <IMG
- WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img329.gif"
- ALT="$ \alpha$">.
- The interpolation operator <IMG
- WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img330.gif"
- ALT="$ \Pi_h^k$"> is defined for sufficiently smooth vector fields, namely for all <!-- MATH
- $\underline v\in H^r(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img331.gif"
- ALT="$ \underline v\in H^r(\mathop{\rm curl})$">
- for any <!-- MATH
- $r>\frac{1}{2}$
- -->
-<IMG
- WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img332.gif"
- ALT="$ r>\frac{1}{2}$"> (see [<A
- HREF="node4.html#Alonso-Valli">1</A>], Lemma 5.1., [<A
- HREF="node4.html#Monk">7</A>] and references therein).
-<BR>
-For Nédélec's FEM of first type we state (without proof) the following optimal estimate in the curl-norm:
- </FONT><P>
-<DIV><B>T<SMALL>HEOREM</SMALL> 5</B>
-If <!-- MATH
- $\mathcal{T}_h$
- -->
-<IMG
- WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img333.gif"
- ALT="$ \mathcal{T}_h$">, <IMG
- WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img334.gif"
- ALT="$ h>0$">, is a regular family of triangulations on <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $"> and <!-- MATH
- $r>\frac{1}{2}$
- -->
-<IMG
- WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img332.gif"
- ALT="$ r>\frac{1}{2}$">, then there exists a constant <IMG
- WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img335.gif"
- ALT="$ C>0$">,
- depending on <IMG
- WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img336.gif"
- ALT="$ r$"> but not on <IMG
- WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img337.gif"
- ALT="$ h$"> or <!-- MATH
- $\underline v$
- -->
-<IMG
- WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ \underline v$">, such that
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_curl-convergence"></A><!-- MATH
- \begin{equation}
-\| \underline v - \Pi_h^k \underline v\|_{H(\mathop{\rm curl}; \Omega )} \leq C\,h^{\min\{r,k\}} \|\underline v\|_{H^r(\mathop{\rm curl};\Omega )}\,,
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="348" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
- SRC="img338.gif"
- ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{H(\mathop{\rm cu...
-... C\,h^{\min\{r,k\}} \Vert\underline v\Vert _{H^r(\mathop{\rm curl};\Omega )}\,,$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(28)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-for all <!-- MATH
- $\underline v\in H^r(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img339.gif"
- ALT="$ \underline v\in H^r(\mathop{\rm curl};\Omega )$">.
- </DIV><P></P>
-<FONT SIZE="-1">
- The result in (<A HREF="node2.html#eq:_curl-convergence">28</A>) was obtained by Alonso and Valli in [<A
- HREF="node4.html#Alonso-Valli">1</A>], extending earlier interpolation
- results by Nédélec in [<A
- HREF="node4.html#Ned1">8</A>] and Monk in [<A
- HREF="node4.html#Monk_92">6</A>].
-<BR>
-Optimal convergence in the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-norm for the error of the FE-approximation of the model
- problem (<A HREF="node1.html#eq:_varform">3</A>) by Nédélec's elements of first type follows from (<A HREF="node2.html#eq:_curl-convergence">28</A>) by Céa's lemma.
- This result has been verified in numerical experiments with a <TT>MATLAB</TT> code, which uses lowest
- order Nédélec elements on affine triangular meshes for 2d problems, as well as with a <TT>deal.II</TT> code, which uses lowest order
- Nédélec elements on bilinear resp. trilinear meshes for 2d resp. 3d problems.
-<BR>
-As for the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-approximation properties of FE spaces based on <!-- MATH
- $\mathcal{R}^k$
- -->
-<IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$ \mathcal{R}^k$"> or <!-- MATH
- $\mathcal{P}^k$
- -->
-<IMG
- WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img228.gif"
- ALT="$ \mathcal{P}^k$">, we could hope for a better order
- than <!-- MATH
- $\mathcal{O}(h^k)$
- -->
-<IMG
- WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img340.gif"
- ALT="$ \mathcal{O}(h^k)$"> at first sight: still, we have <!-- MATH
- $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$
- -->
-<IMG
- WIDTH="165" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img341.gif"
- ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K)$">. However, Nédélec shows in [<A
- HREF="node4.html#Ned1">8</A>]
- that only suboptimality can be expected:
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_L2-approx"></A><!-- MATH
- \begin{equation}
-\| \underline v - \Pi_h^k \underline v\|_{L^2(\Omega )} \leq C h^k |\underline v|_{H^k(\Omega )}\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="244" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img342.gif"
- ALT="$\displaystyle \Vert \underline v - \Pi_h^k \underline v\Vert _{L^2(\Omega )} \leq C h^k \vert\underline v\vert _{H^k(\Omega )}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(29)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- Nédélec uses a standard scaling and Bramble-Hilbert argument to derive (<A HREF="node2.html#eq:_L2-approx">29</A>). Since
- <!-- MATH
- $[\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$
- -->
-<IMG
- WIDTH="254" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img343.gif"
- ALT="$ [\mathbb{P}^{k-1}(K)]^d \subseteq \mathcal{R}^k(K) \subsetneq [\mathbb{P}^k(K)]^d$">, the Bramble-Hilbert argument only guarantees an elementwise
- approximation of order <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> of <IMG
- WIDTH="60" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img344.gif"
- ALT="$ H^k(K)$">-functions from the space <!-- MATH
- $\mathcal{R}^k(K)$
- -->
-<IMG
- WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img345.gif"
- ALT="$ \mathcal{R}^k(K)$">.
-<BR>
-However, in a recent paper Hiptmair uses a duality technique to state optimal convergence of the
- <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-error <!-- MATH
- $\| \underline u - \underline u_h\|_{L^2(\Omega )}$
- -->
-<IMG
- WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img346.gif"
- ALT="$ \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )}$">
- for the 3d case and Nédélec's elements of first type of order <IMG
- WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$ k$"> on tetrahedral meshes
- (see Section 5.3, Theorem 5.8 in [<A
- HREF="node4.html#Hipt">5</A>]):
- </FONT><P>
-<DIV><A NAME="th:_Hiptmair"><B>T<SMALL>HEOREM</SMALL> 6</B></A>
-There is an <!-- MATH
- $s>\frac{1}{2}$
- -->
-<IMG
- WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img347.gif"
- ALT="$ s>\frac{1}{2}$"> such that
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_L2-convergence"></A><!-- MATH
- \begin{equation}
-\| \underline u - \underline u_h\|_{L^2(\Omega )} \leq C h^s \| \underline u - \underline u_h\|_{H(\mathop{\rm curl}; \Omega )}\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="299" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img348.gif"
- ALT="$\displaystyle \Vert \underline u - \underline u_h\Vert _{L^2(\Omega )} \leq C h^s \Vert \underline u - \underline u_h\Vert _{H(\mathop{\rm curl}; \Omega )}\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(30)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-Under the assumption that the boundary <!-- MATH
- $\partial \Omega$
- -->
-<IMG
- WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img9.gif"
- ALT="$ \partial \Omega $"> is smooth or convex, <IMG
- WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img349.gif"
- ALT="$ s=1$"> can be chosen.
- </DIV><P></P>
-<FONT SIZE="-1">
- Several key arguments of the proof in [<A
- HREF="node4.html#Hipt">5</A>] make explicitely use of features that are limited to 3d problems and the
- family of finite elements based on tetrahedrons.
- They cannot be modified trivially to apply to 2d problems or 3d problems on hexahedral meshes. Even worse, it is suggested by the results of
- numerical experiments that one cannot hope to obtain a result similar to (<A HREF="node2.html#eq:_L2-convergence">30</A>).
-<BR>
-A possibility to overcome this deficiency of convergence is to use Nédélec elements of second type, where the full <!-- MATH
- $[\mathbb{P}_k]^d$
- -->
-<IMG
- WIDTH="43" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img350.gif"
- ALT="$ [\mathbb{P}_k]^d$">
- are used as polynomial spaces (see [<A
- HREF="node4.html#Ned2">10</A>]).
-</FONT>
-<P>
-<FONT SIZE="-1"></FONT>
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
+++ /dev/null
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-* with significant contributions from:
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-<HTML>
-<HEAD>
-<TITLE>3 Numerical results</TITLE>
-<META NAME="description" CONTENT="3 Numerical results">
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-<H1><A NAME="SECTION00030000000000000000">
-3 <FONT SIZE="+1">Numerical results</FONT></A>
-</H1><FONT SIZE="-1">
- The numerical results in this section provide some samples of the quality of the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming FEM
- with Nédélec elements of first type and lowest order (polynomial degree <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$">).
-<BR>
-We considered the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in <!-- MATH
- $\Omega = [-1,1]^d$
- -->
-<IMG
- WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img351.gif"
- ALT="$ \Omega = [-1,1]^d$">, <IMG
- WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$ d=2,3$">,
- with homogeneous Dirichlet boundary condition (<A HREF="node1.html#eq:_PCB">2</A>).
-<BR>
-The first few results for the two-dimensional problem have been obtained by a <TT>MATLAB</TT> code.
- For the first example we used the data
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_numerical_ex.1"></A><!-- MATH
- \begin{equation}
-c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\3 - x^2 \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="265" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img352.gif"
- ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = \left(\begin{array}{cc} 3 - y^2 \\ 3 - x^2 \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(31)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- For the second example we have followed the outlines from Appendix A and taken the data from example <A HREF="node5.html#ex:_solutions_from_Laplace">5</A>
- </FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_numerical_ex.2"></A><!-- MATH
- \begin{equation}
-c\equiv 1 \,, \qquad
- \underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,.
-
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="401" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img353.gif"
- ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y) = (2\pi^2 + 1)\left(\begin{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(32)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
- The finite element solution has been computed using Nédélec elements of first type and of polynomial degree <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$"> on a family of
- affine triangular grids.
- The initial coarse grid consisted of <IMG
- WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img354.gif"
- ALT="$ 2^5$"> triangles. The finest grid with <IMG
- WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img355.gif"
- ALT="$ 2^{13}$"> triangles results after five global refinements.
-<BR>
-In Table <A HREF="#table:_matlab">1</A> we see that for both examples we have
- optimal convergence in the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-semiorm, as we would expect from the theoretical results of the
- previous section. As for the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm, it appears that in both examples the convergence
- of the numerical solution is not optimal for our choice of finite elements. In the case of Nédélec elements of first
- type and of polynomial degree <IMG
- WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img108.gif"
- ALT="$ k=1$">, we got only <!-- MATH
- $\mathcal{O}(h)$
- -->
-<IMG
- WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img356.gif"
- ALT="$ \mathcal{O}(h)$">-convergence of the <IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img357.gif"
- ALT="$ L^2$">-error. However, this order of
- convergence is consistent with the result (<A HREF="node2.html#eq:_L2-approx">29</A>) obtained by Nédélec in [<A
- HREF="node4.html#Ned1">8</A>].
-<BR></FONT>
-<P>
-<BR><P></P>
-<DIV ALIGN="CENTER"><A NAME="4142"></A>
-<TABLE>
-<A NAME="table:_matlab"></A>
-<CAPTION><STRONG>Table 1:</STRONG>
-Errors and convergence rates in the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm and <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img359.gif"
- ALT="$ H(\mathop{\rm curl};\Omega )$">-seminorm for the two <TT>MATLAB</TT> examples.</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">grid</TD>
-<TD ALIGN="RIGHT"><IMG
- WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img358.gif"
- ALT="$ \char93 $"> cells</TD>
-<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
- $H(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img46.gif"
- ALT="$ H(\mathop{\rm curl})$">-error</TD>
-<TD ALIGN="CENTER" COLSPAN=2><IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img357.gif"
- ALT="$ L^2$">-error</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">1</TD>
-<TD ALIGN="RIGHT">32</TD>
-<TD ALIGN="CENTER">6.66e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-<TD ALIGN="CENTER">4.66e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">2</TD>
-<TD ALIGN="RIGHT">128</TD>
-<TD ALIGN="CENTER">3.33e-01</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">2.35e-01</TD>
-<TD ALIGN="CENTER">0.98</TD>
-</TR>
-<TR><TD ALIGN="CENTER">example 1</TD>
-<TD ALIGN="CENTER">3</TD>
-<TD ALIGN="RIGHT">512</TD>
-<TD ALIGN="CENTER">1.66e-01</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">1.17e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">4</TD>
-<TD ALIGN="RIGHT">2048</TD>
-<TD ALIGN="CENTER">8.33e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">5.89e-02</TD>
-<TD ALIGN="CENTER">0.99</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">5</TD>
-<TD ALIGN="RIGHT">8192</TD>
-<TD ALIGN="CENTER">4.17e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">2.95e-02</TD>
-<TD ALIGN="CENTER">0.99</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">1</TD>
-<TD ALIGN="RIGHT">32</TD>
-<TD ALIGN="CENTER">3.05e+00</TD>
-<TD ALIGN="CENTER">-</TD>
-<TD ALIGN="CENTER">6.48e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">2</TD>
-<TD ALIGN="RIGHT">128</TD>
-<TD ALIGN="CENTER">1.61e+00</TD>
-<TD ALIGN="CENTER">0.91</TD>
-<TD ALIGN="CENTER">3.22e-01</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-<TR><TD ALIGN="CENTER">example 2</TD>
-<TD ALIGN="CENTER">3</TD>
-<TD ALIGN="RIGHT">512</TD>
-<TD ALIGN="CENTER">0.81e-01</TD>
-<TD ALIGN="CENTER">0.97</TD>
-<TD ALIGN="CENTER">1.60e-01</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">4</TD>
-<TD ALIGN="RIGHT">2048</TD>
-<TD ALIGN="CENTER">0.41e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-<TD ALIGN="CENTER">8.02e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">5</TD>
-<TD ALIGN="RIGHT">8192</TD>
-<TD ALIGN="CENTER">2.05e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-<TD ALIGN="CENTER">4.01e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-</TABLE>
-</DIV>
-</TD></TR>
-</TABLE>
-</DIV><P></P>
-<BR>
-<P>
-
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="fig:conv1"></A><A NAME="4143"></A>
-<TABLE>
-<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
-Convergence of the FE-approximation to the smooth solution of the <TT>MATLAB</TT> example (<A HREF="node3.html#eq:_numerical_ex.1">31</A>)
- in the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm and the <!-- MATH
- $H(\mathop{\rm curl};(\Omega ))$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img361.gif"
- ALT="$ H(\mathop{\rm curl};(\Omega ))$">-seminorm</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<!-- MATH
- $\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}$
- -->
-<IMG
- WIDTH="431" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
- SRC="img360.gif"
- ALT="% latex2html id marker 4265
-\includegraphics[width=9.5cm, height=7cm]{example1_errors.eps}">
-
-</DIV></TD></TR>
-</TABLE>
-</DIV><P></P>
-
-<P>
-
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="fig:conv2"></A><A NAME="4144"></A>
-<TABLE>
-<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
-Convergence of the FE-approximation to the smooth solution of the <TT>MATLAB</TT> example (<A HREF="node3.html#eq:_numerical_ex.2">32</A>)
- in the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm and the <!-- MATH
- $H(\mathop{\rm curl};(\Omega ))$
- -->
-<IMG
- WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img361.gif"
- ALT="$ H(\mathop{\rm curl};(\Omega ))$">-seminorm</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<!-- MATH
- $\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}$
- -->
-<IMG
- WIDTH="431" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
- SRC="img362.gif"
- ALT="% latex2html id marker 4271
-\includegraphics[width=9.5cm, height=7cm]{example2_errors.eps}">
-
-</DIV></TD></TR>
-</TABLE>
-</DIV><P></P>
-
-<P>
-<P>
-<DIV><B>R<SMALL>EMARK</SMALL> 7</B>
-The mesh generation and refinement was done by <TT>PDE-toolbox</TT> commands. Since the <TT>PDE-toolbox</TT> does not support three
- dimensional grids, we restricted ourselves to 2d problems, and we have so far no numerical results for the case of
- tetrahedral grids in 3d.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">As for meshes with quadrilateral cells, numerical results were obtained with a <TT>deal.II</TT> code,
- using the finite element class <TT>fe/fenedelec.cc</TT>. This class provides
- Nédélec's <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming element of first type and lowest order in two and three space dimensions, on
- bilinear quadrilateral, resp. trilinear hexahedral grids. For details about <TT>deal.II</TT>, see [<A
- HREF="node4.html#Deal">2</A>].
- In the following results were obtained for the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in two dimensions using the
- data (<A HREF="node3.html#eq:_numerical_ex.2">32</A>).
- We computed the solution on five successive
- non-affine bilinear grids ( figure <A HREF="node3.html#fig:grid">3</A>), each of which was obtained by global refinement of the previous one.
-</FONT>
-<P>
-
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="fig:grid"></A><A NAME="4146"></A>
-<TABLE>
-<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
-Non-affine bilinear grid used in the <TT>deal.II</TT> code, after one refinement step.</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<!-- MATH
- $\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}$
- -->
-<IMG
- WIDTH="249" HEIGHT="249" ALIGN="BOTTOM" BORDER="0"
- SRC="img363.gif"
- ALT="\includegraphics[width=5.5cm, height=5.5cm]{grid.eps}">
-
-</DIV></TD></TR>
-</TABLE>
-</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">Again, in Table <A HREF="#table:_2d-deal">2</A> we can observe optimal convergence of order <!-- MATH
- $\mathcal{O}(h)$
- -->
-<IMG
- WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img356.gif"
- ALT="$ \mathcal{O}(h)$"> in the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-norm.
-The same order of convergence is obtained for the error in the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm.
-<BR></FONT>
-<P>
-<BR><P></P>
-<DIV ALIGN="CENTER"><A NAME="4147"></A>
-<TABLE>
-<a name="table:_2d-deal"></a>
-<CAPTION><STRONG>Table 2:</STRONG>
-Errors and convergence rates in the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img359.gif"
- ALT="$ H(\mathop{\rm curl};\Omega )$">- and <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm for the 2d-example solved with <TT>deal.II</TT>.
-</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
-<TR><TD ALIGN="CENTER">grid</TD>
-<TD ALIGN="RIGHT"><IMG
- WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img358.gif"
- ALT="$ \char93 $"> cells</TD>
-<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
- $H(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img46.gif"
- ALT="$ H(\mathop{\rm curl})$">-error</TD>
-<TD ALIGN="CENTER" COLSPAN=2><IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img357.gif"
- ALT="$ L^2$">-error</TD>
-</TR>
-<TR><TD ALIGN="CENTER">1</TD>
-<TD ALIGN="RIGHT">4</TD>
-<TD ALIGN="CENTER">6.112e+00</TD>
-<TD ALIGN="CENTER">-</TD>
-<TD ALIGN="CENTER">1.442e+00</TD>
-<TD ALIGN="CENTER">-</TD>
-</TR>
-<TR><TD ALIGN="CENTER">2</TD>
-<TD ALIGN="RIGHT">16</TD>
-<TD ALIGN="CENTER">3.688e+00</TD>
-<TD ALIGN="CENTER">0.73</TD>
-<TD ALIGN="CENTER">6.765e-01</TD>
-<TD ALIGN="CENTER">1.09</TD>
-</TR>
-<TR><TD ALIGN="CENTER">3</TD>
-<TD ALIGN="RIGHT">64</TD>
-<TD ALIGN="CENTER">1.991e+00</TD>
-<TD ALIGN="CENTER">0.89</TD>
-<TD ALIGN="CENTER">3.280e-01</TD>
-<TD ALIGN="CENTER">1.04</TD>
-</TR>
-<TR><TD ALIGN="CENTER">4</TD>
-<TD ALIGN="RIGHT">256</TD>
-<TD ALIGN="CENTER">1.015e+00</TD>
-<TD ALIGN="CENTER">0.97</TD>
-<TD ALIGN="CENTER">1.617e-01</TD>
-<TD ALIGN="CENTER">1.02</TD>
-</TR>
-<TR><TD ALIGN="CENTER">5</TD>
-<TD ALIGN="RIGHT">1024</TD>
-<TD ALIGN="CENTER">5.098e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-<TD ALIGN="CENTER">8.049e-02</TD>
-<TD ALIGN="CENTER">1.01</TD>
-</TR>
-</TABLE>
-</DIV>
-</TD></TR>
-</TABLE>
-</DIV><P></P>
-<BR>
-<P>
-<FONT SIZE="-1">With <TT>deal.II</TT>, we are also able to treat 3d problems on hexahedral grids. For our type of problem,
-Nédélec's <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">-conforming elements of first type and lowest order, based on a cubic reference element, are available.
-<BR>
-We computed an approximation to the model problem (<A HREF="node1.html#eq:model_problem">1</A>) in 3d using the data
- </FONT><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation}
-c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{ccc} xy(1 - y^2)(1-z^2) + 2xy(1-z^2) \\
- y^2(1 - x^2)(1-z^2) + (1-y^2)(2-x^2-z^2) \\
- yz(1 - x^2)(1-y^2) + 2yz(1-x^2)
- \end{array}\right)\,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="557" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
- SRC="img364.gif"
- ALT="$\displaystyle c\equiv 1 \,, \qquad \underline f(x,y,z) = \left(\begin{array}{cc...
-... + (1-y^2)(2-x^2-z^2) \\ yz(1 - x^2)(1-y^2) + 2yz(1-x^2) \end{array}\right)\,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(33)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-In a first experiment, the finite element solution was computed on five successive globally refined affine grids. In a second
-computation, we approximated the solution of the same problem on five successive globally refined non-affine trilinear grids.
-<BR>
-We see in Table <A HREF="#table:_3d-deal">3</A> that in both cases we observe again convergence of order <!-- MATH
- $\mathcal{O}(h)$
- -->
-<IMG
- WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img356.gif"
- ALT="$ \mathcal{O}(h)$"> in the
-<!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$ H(\mathop {\rm curl};\Omega )$">- and the <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm.
-<BR></FONT>
-<P>
-<BR><P></P>
-<DIV ALIGN="CENTER"><A NAME="4148"></A>
-<TABLE>
-<a name="table:_3d-deal"></A>
-<CAPTION><STRONG>Table 3:</STRONG>
-Errors and convergence rates in the <!-- MATH
- $H(\mathop{\rm curl};\Omega )$
- -->
-<IMG
- WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img359.gif"
- ALT="$ H(\mathop{\rm curl};\Omega )$">- and <!-- MATH
- $L^2(\Omega )$
- -->
-<IMG
- WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ L^2(\Omega )$">-norm for the 3d-example solved with <TT>deal.II</TT>. The first data
-set is for the computation on a family of affine grids, the second set of data is for non-affine trilinear grids.</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">grid</TD>
-<TD ALIGN="RIGHT"><IMG
- WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img358.gif"
- ALT="$ \char93 $"> cells</TD>
-<TD ALIGN="CENTER" COLSPAN=2><!-- MATH
- $H(\mathop{\rm curl})$
- -->
-<IMG
- WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img46.gif"
- ALT="$ H(\mathop{\rm curl})$">-error</TD>
-<TD ALIGN="CENTER" COLSPAN=2><IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img357.gif"
- ALT="$ L^2$">-error</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">1</TD>
-<TD ALIGN="RIGHT">8</TD>
-<TD ALIGN="CENTER">7.696e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-<TD ALIGN="CENTER">6.609e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">2</TD>
-<TD ALIGN="RIGHT">64</TD>
-<TD ALIGN="CENTER">4.088e-01</TD>
-<TD ALIGN="CENTER">0.91</TD>
-<TD ALIGN="CENTER">2.943e-01</TD>
-<TD ALIGN="CENTER">1.17</TD>
-</TR>
-<TR><TD ALIGN="CENTER">affine grids</TD>
-<TD ALIGN="CENTER">3</TD>
-<TD ALIGN="RIGHT">512</TD>
-<TD ALIGN="CENTER">2.075e-01</TD>
-<TD ALIGN="CENTER">0.98</TD>
-<TD ALIGN="CENTER">1.408e-01</TD>
-<TD ALIGN="CENTER">1.06</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">4</TD>
-<TD ALIGN="RIGHT">4096</TD>
-<TD ALIGN="CENTER">1.041e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-<TD ALIGN="CENTER">6.955e-02</TD>
-<TD ALIGN="CENTER">1.02</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">5</TD>
-<TD ALIGN="RIGHT">32768</TD>
-<TD ALIGN="CENTER">5.210e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">3.467e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">1</TD>
-<TD ALIGN="RIGHT">8</TD>
-<TD ALIGN="CENTER">7.716e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-<TD ALIGN="CENTER">6.611e-01</TD>
-<TD ALIGN="CENTER">-</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">2</TD>
-<TD ALIGN="RIGHT">64</TD>
-<TD ALIGN="CENTER">4.108e-01</TD>
-<TD ALIGN="CENTER">0.91</TD>
-<TD ALIGN="CENTER">2.955e-01</TD>
-<TD ALIGN="CENTER">1.16</TD>
-</TR>
-<TR><TD ALIGN="CENTER">non-affine grids</TD>
-<TD ALIGN="CENTER">3</TD>
-<TD ALIGN="RIGHT">512</TD>
-<TD ALIGN="CENTER">2.085e-01</TD>
-<TD ALIGN="CENTER">0.98</TD>
-<TD ALIGN="CENTER">1.413e-01</TD>
-<TD ALIGN="CENTER">1.06</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">4</TD>
-<TD ALIGN="RIGHT">4096</TD>
-<TD ALIGN="CENTER">1.046e-01</TD>
-<TD ALIGN="CENTER">0.99</TD>
-<TD ALIGN="CENTER">6.982e-02</TD>
-<TD ALIGN="CENTER">1.02</TD>
-</TR>
-<TR><TD ALIGN="CENTER"> </TD>
-<TD ALIGN="CENTER">5</TD>
-<TD ALIGN="RIGHT">32768</TD>
-<TD ALIGN="CENTER">5.237e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-<TD ALIGN="CENTER">3.480e-02</TD>
-<TD ALIGN="CENTER">1.00</TD>
-</TR>
-</TABLE>
-</DIV>
-</TD></TR>
-</TABLE>
-</DIV><P></P>
-<BR><FONT SIZE="-1">
-The conclusion that can be drawn from these numerical experiments is, that the restriction to three-dimensional tetrahedral grids
-of Hiptmair's result on the <IMG
- WIDTH="26" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img357.gif"
- ALT="$ L^2$">-convergence of the error (<A HREF="node2.html#th:_Hiptmair">6</A>) cannot be relaxed.
-<BR></FONT>
-<P>
-<FONT SIZE="-1">Finally, here are some pretty pictures: the vector field plots from the <TT>MATLAB</TT> computations.
-</FONT>
-<P>
-
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="fig:field1"></A><A NAME="4149"></A>
-<TABLE>
-<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
-Vector-field plot of the FE-solution of example (<A HREF="node3.html#eq:_numerical_ex.1">31</A>).</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<!-- MATH
- $\includegraphics[width=9.5cm, height=7cm]{field1.eps}$
- -->
-<IMG
- WIDTH="432" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
- SRC="img365.gif"
- ALT="\includegraphics[width=9.5cm, height=7cm]{field1.eps}">
-
-</DIV></TD></TR>
-</TABLE>
-</DIV><P></P>
-
-<P>
-
-<P></P>
-<DIV ALIGN="CENTER"><A NAME="fig:field2"></A><A NAME="4150"></A>
-<TABLE>
-<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
-Vector-field plot of the FE-solution of example (<A HREF="node3.html#eq:_numerical_ex.2">32</A>).</CAPTION>
-<TR><TD>
-<DIV ALIGN="CENTER">
-<!-- MATH
- $\includegraphics[width=9.5cm, height=7cm]{field2.eps}$
- -->
-<IMG
- WIDTH="432" HEIGHT="318" ALIGN="BOTTOM" BORDER="0"
- SRC="img366.gif"
- ALT="\includegraphics[width=9.5cm, height=7cm]{field2.eps}">
-
-</DIV></TD></TR>
-</TABLE>
-</DIV><P></P>
-<FONT SIZE="-1">
-</FONT>
-<P>
-
-<P>
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
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-<HTML>
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-
-<H2><A NAME="SECTION00040000000000000000">
-Bibliography</A>
-</H2><DL COMPACT><DD><P></P><DT><A NAME="Alonso-Valli">1</A>
-<DD>
-A. Alonso and A. Valli.
-<BR>An optimal domain decomposition preconditioner for low-frequency
- time-harmonic Maxwell equations.
-<BR><EM>Math. Comp.</EM>, 68(226):607-631, 1999.
-
-<P></P><DT><A NAME="Deal">2</A>
-<DD>
-W. Bangerth, R. Hartmann, and G. Kanschat.
-<BR><EM><TT>deal.II</TT> Differential Equations Analysis Library,
- Technical Reference</EM>.
-<BR>IWR, Universität Heidelberg.
-<BR><TT>http://www.dealii.org</TT>.
-
-<P></P><DT><A NAME="Brezzi-Fortin">3</A>
-<DD>
-F. Brezzi and M. Fortin.
-<BR><EM>Mixed and Hybrid Finite Element Methods</EM>, volume 15 of <EM> Springer Series in Computational Mathematics</EM>.
-<BR>Springer-Verlag, New York, 1991.
-
-<P></P><DT><A NAME="Girault-Raviart">4</A>
-<DD>
-V. Girault and P.-A. Raviart.
-<BR><EM>Finite Element Approximation of the Navier-Stokes Equations</EM>,
- volume 749 of <EM>Lecture Notes in Mathematics</EM>.
-<BR>Springer-Verlag, Berlin, Heidelberg, 1979, 1981.
-
-<P></P><DT><A NAME="Hipt">5</A>
-<DD>
-R. Hiptmair.
-<BR>Finite elements in computational electromagnetism.
-<BR>In <EM>Acta Numerica</EM>, pages 1-103. Cambridge University press,
- 2002.
-
-<P></P><DT><A NAME="Monk_92">6</A>
-<DD>
-P. Monk.
-<BR>Analysis of a finite element method for Maxwell's equations.
-<BR><EM>SIAM J. Numer. Anal</EM>, 29:714-729, 1992.
-
-<P></P><DT><A NAME="Monk">7</A>
-<DD>
-P. Monk.
-<BR>A simple proof for an edge element discretization of Maxwell's
- equations.
-<BR>Submitted for publication. Download version available on Monk's
- webpage: www.math.udel.edu./ monk, 2001.
-
-<P></P><DT><A NAME="Ned1">8</A>
-<DD>
-J. C. Nédélec.
-<BR>Mixed finite elements in <!-- MATH
- $\mathbb{R}^3$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img367.gif"
- ALT="$ \mathbb{R}^3$">.
-<BR><EM>Numer. Math.</EM>, 35:315-341, 1980.
-
-<P></P><DT><A NAME="Ned3">9</A>
-<DD>
-J. C. Nédélec.
-<BR>Elements finis mixtes incompressibles pour l'équation de Stokes
- dans <!-- MATH
- $\mathbb{R}^3$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img367.gif"
- ALT="$ \mathbb{R}^3$">.
-<BR><EM>Numer. Math.</EM>, 39:97-112, 1982.
-
-<P></P><DT><A NAME="Ned2">10</A>
-<DD>
-J. C. Nédélec.
-<BR>A new family of mixed finite elements in <!-- MATH
- $\mathbb{R}^3$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img367.gif"
- ALT="$ \mathbb{R}^3$">.
-<BR><EM>Numer. Math.</EM>, 50:57-81, 1986.
-
-<P></P><DT><A NAME="Demko">11</A>
-<DD>
-W. Rachowicz and L. Demkowicz.
-<BR>A two-dimensional hp-adaptive finite element package for
- electromagnetics (2Dhp90_EM).
-<BR>Ticam Report 98-16, TICAM, 1998.
-<BR>Download version available on Demkowicz' webpage:
- www.ices.utexas.edu/ Leszek.
-
-<P></P><DT><A NAME="Demko3d">12</A>
-<DD>
-W. Rachowicz and L. Demkowicz.
-<BR>A three-dimensional hp-adaptive finite element package for
- electromagnetics (3Dhp90_EM).
-<BR>Ticam Report 00-04.2000, TICAM, 2000.
-<BR>Download version available on Demkowicz' webpage:
- www.ices.utexas.edu/ Leszek.
-</DL>
-
-</FONT><BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
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-
-<H1><A NAME="SECTION00050000000000000000">
-A. <FONT SIZE="+1">Construction of solutions in 2d</FONT></A>
-</H1>
-<P>
-<FONT SIZE="-1">We present how divergence-free solutions of the model problem (<A HREF="node1.html#eq:model_problem">1</A>) on a domain <!-- MATH
- $\Omega \subset\mathbb{R}^2$
- -->
-<IMG
- WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img167.gif"
- ALT="$ \Omega \subset\mathbb{R}^2$"> with perfectly
- conducting boundary
- can be constructed from solutions of the scalar Laplace equation.
-<BR></FONT>
-<P>
-<P>
-<DIV><B>P<SMALL>ROPOSITION</SMALL> 9</B>
-Let <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $"> be a sufficiently smooth domain in <!-- MATH
- $\mathbb{R}^2$
- -->
-<IMG
- WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img368.gif"
- ALT="$ \mathbb{R}^2$">, <!-- MATH
- $\varphi (x,y)$
- -->
-<IMG
- WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img369.gif"
- ALT="$ \varphi (x,y)$"> a sufficently smooth scalar function on <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $"> and the coefficient <IMG
- WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img370.gif"
- ALT="$ c>0$">
- globally constant.
-<BR>
-Let <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img371.gif"
- ALT="$ w$"> be a solution of the scalar equation
- <P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_Laplace"></A><!-- MATH
- \begin{equation}
-\begin{split}
- -\Delta w + c\, w &= \varphi \quad \mathrm{in} \quad \Omega \\
- \underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,.
- \end{split}
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="382" HEIGHT="55" BORDER="0"
- SRC="img372.gif"
- ALT="\begin{displaymath}\begin{split}-\Delta w + c\, w &= \varphi \quad \mathrm{in} \...
-... w &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(34)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-Then, <!-- MATH
- $\underline E := \nabla^{\perp} w$
- -->
-<IMG
- WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img373.gif"
- ALT="$ \underline E := \nabla^{\perp} w$"> is a solution of the model equation
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- \underline \mathop{\rm curl}\mathop{\rm curl}\underline E + c\, \underline E = \underline f \quad \mathrm{in} \quad \Omega \,, \\
- \underline E \wedge \underline n = 0 \quad \mathrm{on} \quad \partial \Omega \,,
- \end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="397" HEIGHT="60" BORDER="0"
- SRC="img374.gif"
- ALT="\begin{displaymath}\begin{split}\underline \mathop{\rm curl}\mathop{\rm curl}\un...
-...e n = 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-with right hand side <!-- MATH
- $\underline f := \nabla^{\perp} \varphi$
- -->
-<IMG
- WIDTH="82" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img375.gif"
- ALT="$ \underline f := \nabla^{\perp} \varphi $">.
-<BR>
-We use the notation <!-- MATH
- $\nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\-\partial _x\varphi
-\end{array}\right)$
- -->
-<IMG
- WIDTH="220" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img376.gif"
- ALT="$ \nabla^{\perp} \varphi := \boldsymbol{R}\nabla\varphi = \left(\begin{array}{cc} \partial _y\varphi \\ -\partial _x\varphi
-\end{array}\right)$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">P<SMALL>ROOF. </SMALL>
- We first show the correspondence of the boundary conditions. With the definition <!-- MATH
- $\underline E := \nabla^{\perp} w$
- -->
-<IMG
- WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img373.gif"
- ALT="$ \underline E := \nabla^{\perp} w$"> it holds
- </FONT><!-- MATH
- \begin{displaymath}
-\underline E \wedge \underline n = \underline E \cdot \underline t = {\nabla w}^T \boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="303" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
- SRC="img377.gif"
- ALT="$\displaystyle \underline E \wedge \underline n = \underline E \cdot \underline ...
-...boldsymbol{R}^T\boldsymbol{R}\, \underline n = \nabla w \cdot \underline n \,.
-$">
-</DIV><P></P><FONT SIZE="-1">
- It remains to show that <!-- MATH
- $\underline E$
- -->
-<IMG
- WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img378.gif"
- ALT="$ \underline E$"> solves the model problem for an appropriate right hand side.
- First, note that <!-- MATH
- $\underline E$
- -->
-<IMG
- WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img378.gif"
- ALT="$ \underline E$"> is divergence-free:
- <!-- MATH
- $\nabla\cdot\nabla^{\perp}w = 0$
- -->
-<IMG
- WIDTH="104" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img379.gif"
- ALT="$ \nabla\cdot\nabla^{\perp}w = 0$"> for all <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img371.gif"
- ALT="$ w$">. Hence, the identity <!-- MATH
- $\underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$
- -->
-<IMG
- WIDTH="229" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img380.gif"
- ALT="$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = \nabla(\nabla\cdot\underline E) - \Delta\underline E$">
- reduces to <!-- MATH
- $\underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$
- -->
-<IMG
- WIDTH="152" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img381.gif"
- ALT="$ \underline \mathop{\rm curl}\mathop{\rm curl}\underline E = - \Delta\underline E$">. The observation that for smooth data <!-- MATH
- $\nabla^{\perp}w$
- -->
-<IMG
- WIDTH="45" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img382.gif"
- ALT="$ \nabla^{\perp}w$"> solves the Laplace equation
- (<A HREF="node5.html#eq:_Laplace">34</A>) with right hand side <!-- MATH
- $\nabla^{\perp} \varphi$
- -->
-<IMG
- WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img383.gif"
- ALT="$ \nabla^{\perp} \varphi $"> concludes the proof.
-
-</FONT>
-<P>
-<P>
-<DIV><A NAME="ex:_solutions_from_Laplace"><B>E<SMALL>XAMPLE</SMALL> 5</B></A> (Solutions from eigenfunctions of the Laplacian)
-Choose <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img371.gif"
- ALT="$ w$"> to be a solution of the eigenvalue problem
- <P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- -\Delta w &= \lambda \, w \quad \mathrm{in} \quad \Omega \\
- \underline n \cdot \nabla w &= 0 \quad \mathrm{on} \quad \partial \Omega \,,
- \end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="364" HEIGHT="55" BORDER="0"
- SRC="img384.gif"
- ALT="\begin{displaymath}\begin{split}-\Delta w &= \lambda \, w \quad \mathrm{in} \qua...
-... w &= 0 \quad \mathrm{on} \quad \partial \Omega \,, \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-and set <!-- MATH
- $\varphi = (\lambda + c)\,w$
- -->
-<IMG
- WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img385.gif"
- ALT="$ \varphi = (\lambda + c)\,w$">.
-<BR>
-As an example, take <!-- MATH
- $\Omega = [-1,1]^2$
- -->
-<IMG
- WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img386.gif"
- ALT="$ \Omega = [-1,1]^2$"> and <!-- MATH
- $\lambda = 2\pi^2$
- -->
-<IMG
- WIDTH="67" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
- SRC="img387.gif"
- ALT="$ \lambda = 2\pi^2$">. Then, <!-- MATH
- $w = \cos\pi x\cos\pi y$
- -->
-<IMG
- WIDTH="138" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img388.gif"
- ALT="$ w = \cos\pi x\cos\pi y$"> is an eigenfunction and we compute
- <!-- MATH
- \begin{displaymath}
-\underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,, \qquad
- \underline E = \pi \left(\begin{array}{cc} \cos\pi x\sin\pi y \\-\sin\pi x\cos\pi y \end{array}\right)\,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="536" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img389.gif"
- ALT="$\displaystyle \underline f = (2\pi^2 + c)\pi \left(\begin{array}{cc} \cos\pi x\...
-...in{array}{cc} \cos\pi x\sin\pi y \\ -\sin\pi x\cos\pi y \end{array}\right)\,.
-$">
-</DIV><P></P></DIV><P></P>
-
-<P>
-<P>
-<DIV><B>E<SMALL>XAMPLE</SMALL> 6</B> (Solutions from any scalar function satiesfying the boundary condition)
-Take again <!-- MATH
- $\Omega = [-1,1]^2$
- -->
-<IMG
- WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img386.gif"
- ALT="$ \Omega = [-1,1]^2$">. We have to find a scalar function <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img371.gif"
- ALT="$ w$"> which satiesfies the homogeneous Neumann boundary condition. Take
- for example <!-- MATH
- $w(x,y) = (1-x^2)^2(1-y^2)^2$
- -->
-<IMG
- WIDTH="220" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img390.gif"
- ALT="$ w(x,y) = (1-x^2)^2(1-y^2)^2$">, for which we have <!-- MATH
- $\underline n \cdot \nabla w = 0$
- -->
-<IMG
- WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img391.gif"
- ALT="$ \underline n \cdot \nabla w = 0$"> on <!-- MATH
- $\partial [-1,1]^2$
- -->
-<IMG
- WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img392.gif"
- ALT="$ \partial [-1,1]^2$">. The right hand side is then
- <!-- MATH
- $\varphi = -\Delta w + c w$
- -->
-<IMG
- WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img393.gif"
- ALT="$ \varphi = -\Delta w + c w$">.
- </DIV><P></P>
-
-<P>
-<FONT SIZE="-1">
-</FONT>
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
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-<HTML>
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-<TITLE>B. Time-harmonic Maxwell's equations with low-frequency approximation</TITLE>
-<META NAME="description" CONTENT="B. Time-harmonic Maxwell's equations with low-frequency approximation">
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-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
-
-<UL>
-<LI><A NAME="tex2html110"
- HREF="#SECTION00061000000000000000">Time-harmonic, low-frequency case</A>
-</UL>
-<!--End of Table of Child-Links-->
-<HR>
-
-<H1><A NAME="SECTION00060000000000000000"></A>
-<A NAME="appendix:_Maxwell_s_Eq."></A>
-<BR>
-B. <FONT SIZE="+1">Time-harmonic Maxwell's equations with low-frequency approximation</FONT>
-</H1><FONT SIZE="-1">
-We show, how the model problem can be derived from the time-harmonic Maxwell's equations in the low-frequency case.
-We follow the outline of [<A
- HREF="node4.html#Alonso-Valli">1</A>]:
-<BR>
-We consider the following primal formulation of Maxwell's equations:
-</FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_maxwell"></A><!-- MATH
- \begin{equation}
-\begin{split}
-\varepsilon \frac{\partial \mathcal{E}}{\partial t} & = \mathop{\rm curl}\mathcal{H} - \sigma \mathcal{E} \,, \\
- \mu \frac{\partial \mathcal{H}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,,
-\end{split}
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="361" HEIGHT="85" BORDER="0"
- SRC="img394.gif"
- ALT="\begin{displaymath}\begin{split}\varepsilon \frac{\partial \mathcal{E}}{\partial...
-...}}{\partial t} & = -\mathop{\rm curl}\mathcal{E}\,, \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(35)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-where <!-- MATH
- $\mathcal{E}$
- -->
-<IMG
- WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img395.gif"
- ALT="$ \mathcal{E}$"> and <!-- MATH
- $\mathcal{H}$
- -->
-<IMG
- WIDTH="21" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img396.gif"
- ALT="$ \mathcal{H}$"> are the electric and magnetic field.
-<!-- MATH
- $\varepsilon (x), \mu(x)$
- -->
-<IMG
- WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img397.gif"
- ALT="$ \varepsilon (x), \mu(x)$"> are the dielectric and magnetic permeability coefficients, and <IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img398.gif"
- ALT="$ \sigma(x)$"> denotes the electric conductivity.
-<!-- MATH
- $\varepsilon (x), \mu(x)$
- -->
-<IMG
- WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img397.gif"
- ALT="$ \varepsilon (x), \mu(x)$"> and <IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img398.gif"
- ALT="$ \sigma(x)$"> are assumed to be symmetric matrices in <!-- MATH
- $L^{\infty}(\Omega )^{d\times d}$
- -->
-<IMG
- WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img399.gif"
- ALT="$ L^{\infty}(\Omega )^{d\times d}$">, and <!-- MATH
- $\varepsilon (x)$
- -->
-<IMG
- WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img400.gif"
- ALT="$ \varepsilon (x)$"> and <IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img401.gif"
- ALT="$ \mu(x)$"> are
-positive definite. <IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img398.gif"
- ALT="$ \sigma(x)$"> is positive definite in a conductor and vanishes in an insulator.
-</FONT>
-<P>
-
-<H2><A NAME="SECTION00061000000000000000">
-Time-harmonic, low-frequency case</A>
-</H2><FONT SIZE="-1">
-We assume that <!-- MATH
- $\mathcal{E}(x,t)$
- -->
-<IMG
- WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img402.gif"
- ALT="$ \mathcal{E}(x,t)$"> and <!-- MATH
- $\mathcal{H}(x,t)$
- -->
-<IMG
- WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img403.gif"
- ALT="$ \mathcal{H}(x,t)$"> are <I>time-harmonic</I>, i. e. they can be represented as
-</FONT><P></P>
-<DIV ALIGN="CENTER"><!-- MATH
- \begin{equation*}
-\begin{split}
- \mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(i\omega t)\right) \,, \\
- \mathcal{H}(x,t) &= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,.
-\end{split}
-\end{equation*}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="396" HEIGHT="55" BORDER="0"
- SRC="img404.gif"
- ALT="\begin{displaymath}\begin{split}\mathcal{E}(x,t) &= \mathrm{Re} \left(E(x) \exp(...
-...= \mathrm{Re} \left(H(x) \exp(i\omega t)\right) \,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-Here, <!-- MATH
- $E(x), H(x)$
- -->
-<IMG
- WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img405.gif"
- ALT="$ E(x), H(x)$"> are complex-valued vector fields and <!-- MATH
- $\omega\neq 0$
- -->
-<IMG
- WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img406.gif"
- ALT="$ \omega\neq 0$"> is a given angular frequency.
-</FONT><P>
-<DIV><B>R<SMALL>EMARK</SMALL> 8</B>
-For example, a monofrequent laser can be described by the time-harmonic Maxwell's equations.</DIV><P></P>
-
-<P>
-<FONT SIZE="-1">In the time-harmonic case the space and time variables decouple and we can eliminate the time dependency. For this, we ask
-<!-- MATH
- $E(x) \exp(i\omega t)$
- -->
-<IMG
- WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img407.gif"
- ALT="$ E(x) \exp(i\omega t)$"> and <!-- MATH
- $H(x) \exp(i\omega t)$
- -->
-<IMG
- WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img408.gif"
- ALT="$ H(x) \exp(i\omega t)$"> to satiesfy (<A HREF="node6.html#eq:_maxwell">35</A>).
-By then inserting the second equation of (<A HREF="node6.html#eq:_maxwell">35</A>) into the first one, we can eliminate the magnetic field <IMG
- WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img409.gif"
- ALT="$ H(x)$">. This yields
-</FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="284" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img410.gif"
- ALT="$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) - \omega^2\varepsilon E + i\omega\sigma E = 0
-$">
-</DIV><P></P><FONT SIZE="-1">
-In the <I>low-frequency case</I> where <IMG
- WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img411.gif"
- ALT="$ \vert\omega\vert$"> is small, it is known that for general materials the material parameters are such that
-</FONT><!-- MATH
- \begin{displaymath}
-\omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,.
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="201" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img412.gif"
- ALT="$\displaystyle \omega^2\varepsilon \ll \mu^{-1} \,,\quad \omega^2\varepsilon \ll \omega\sigma \,.
-$">
-</DIV><P></P><FONT SIZE="-1">
-Hence, neglecting the expression <!-- MATH
- $\omega^2\varepsilon E(x)$
- -->
-<IMG
- WIDTH="71" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
- SRC="img413.gif"
- ALT="$ \omega^2\varepsilon E(x)$"> is reasonable and it brings us to the low-frequency approximation of the
-time-harmonic Maxwell's equations:
-</FONT><!-- MATH
- \begin{displaymath}
-\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0
-\end{displaymath}
- -->
-<P></P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="222" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
- SRC="img414.gif"
- ALT="$\displaystyle \mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}E) + i\omega\sigma E = 0
-$">
-</DIV><P></P>
-<P>
-<FONT SIZE="-1">We consider this equation in a conductor <IMG
- WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
- SRC="img10.gif"
- ALT="$ \Omega $"> (<IMG
- WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
- SRC="img398.gif"
- ALT="$ \sigma(x)$"> pos. def.) and a impose Dirichlet boundary condition on the tangential trace
-of the field:
-</FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_non-hom._bc"></A><!-- MATH
- \begin{equation}
-E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="170" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img415.gif"
- ALT="$\displaystyle E \wedge n = \Phi \quad \mathrm{on} \quad \partial \Omega \,.$"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(36)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-Proceeding as in [<A
- HREF="node4.html#Alonso-Valli">1</A>], we assume that a vector funciton <IMG
- WIDTH="20" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
- SRC="img416.gif"
- ALT="$ \tilde{E}$"> is known,
-satiesfying (<A HREF="node6.html#eq:_non-hom._bc">36</A>), and we end up with the following boundary value problem for <!-- MATH
- $\underline u = E - \tilde{E}$
- -->
-<IMG
- WIDTH="89" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
- SRC="img417.gif"
- ALT="$ \underline u = E - \tilde{E}$">
-</FONT><P></P>
-<DIV ALIGN="CENTER"><A NAME="eq:_real_problem"></A><!-- MATH
- \begin{equation}
-\begin{split}
-\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) + i\omega\sigma u &= F \quad \mathrm{in} \quad \Omega \,, \\
- u \wedge n &= 0 \quad \mathrm{on} \quad \partial \Omega \,.
-\end{split}
-\end{equation}
- -->
-<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE">
-<TD NOWRAP ALIGN="CENTER"><IMG
- WIDTH="426" HEIGHT="55" BORDER="0"
- SRC="img418.gif"
- ALT="\begin{displaymath}\begin{split}\mathop{\rm curl}(\mu^{-1} \mathop{\rm curl}u) +...
-... n &= 0 \quad \mathrm{on} \quad \partial \Omega \,. \end{split}\end{displaymath}"></TD>
-<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
-(37)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P><FONT SIZE="-1">
-Although problem (<A HREF="node6.html#eq:_real_problem">37</A>) is complex-valued, finding a finite element method to approximate (<A HREF="node6.html#eq:_real_problem">37</A>)
-basically boils down to finding a finite element method for the real valued model problem (<A HREF="node1.html#eq:model_problem">1</A>).
-</FONT>
-<P>
-<BR><HR>
-<ADDRESS>
-
-2003-04-30
-</ADDRESS>
-</BODY>
-</HTML>
projects / dealii-svn.git / commitdiff
