Mathematics of Computation

Author(s): Joachim Schöberl.
Journal: Math. Comp. 77 (2008), 633-649.
MSC (2000): Primary 65N30
Posted: December 12, 2007
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Abstract: Maxwell equations are posed as variational boundary value problems in the function space $H(\operatorname{curl})$ and are discretized by Nédélec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

1.: M. Ainsworth and J. Coyle.
Hierarchic finite element bases on unstructured tetrahedral meshes.
Int. J. Num. Meth. Eng., 58(14), 2103-2130. MR 2022172 (2004j:65178)
2.: M. Ainsworth and T. Oden.
A Posteriori Error Estimation in Finite Element Analysis.
Wiley-Interscience, 2000. MR 1885308 (2003b:65001)
3.: D. N. Arnold, R. S. Falk, and R. Winther.
Multigrid in ${H}(\operatorname{div})$ and ${H}(\operatorname{curl})$ .
Numer. Math., 85:197-218, 2000. MR 1754719 (2001d:65161)
4.: R. Beck, R. Hiptmair, R. Hoppe, and B. Wohlmuth.
Residual based a posteriori error estimators for eddy current computations.
MAN, 34(1):159-182, 2000. MR 1735971 (2000k:65203)
5.: R. Beck, R. Hiptmair, and B. Wohlmuth.
Hierarchical error estimator for eddy current computation.
In ENUMATH99: Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applictions (ed. P. Neitaanmäki and T. Tiihonen), 110-120. World Scientific, Singapore, 2000. MR 1936173
6.: D. Boffi.
Discrete compactness and Fortin operator for edge elements.
Numer. Math., 87:229-246, 2000. MR 1804657 (2001k:65168)
7.: D. Boffi.
A note on the discrete compactness property and the de Rham diagram.
Appl. Math. Letters, 14:33-38, 2001. MR 1793699 (2001g:65145)
8.: A. Bossavit.
Mixed finite elements and the complex of Whitney forms.
In J. Whiteman, editor, The Mathematics of Finite Elements and Applications VI, 137-144. Academic Press, London, 1988. MR 956893 (89k:58028)
9.: D. Braess and J. Schöberl.
Equilibrated residual error estimators for Maxwell's equations.
Technical Report 2006-19, Johann Radon Institute for Computational and Applied Mathematics (RICAM), 2006.
10.: C. Carstensen.
A posteriori error estimate for the mixed finite element method.
Math. Comp., 66:465-476, 1997. MR 1408371 (98a:65162)
11.: P. Clément.
Approximation by finite element functions using local regularization.
R.A.I.R.O. Anal. Numer., R2:77-84, 1975. MR 0400739 (53:4569)
12.: S. Cochez and S. Nicaise.
Uniform a posteriori error estimation for the heterogeneous Maxwell equations.
Report LAMAV 06.05, Université de Valenciennes et du Hainaut Cambrésis.
13.: L. Demkowicz and I. Babuška.
Optimal interpolation error estimates for edge finite elements of variable order in .
Technical Report 01-11, TICAM, University of Texas at Austin, 2001.
14.: L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz.
De Rham diagram for finite element spaces.
Comput. Math. Appl., 39(7-8):29-38, 2000. MR 1746160 (2000m:78052)
15.: V. Girault and P. A. Raviart.
Finite Element Methods for Navier-Stokes Equations.
Springer, Berlin, Heidelberg, New York, 1986. MR 851383 (88b:65129)
16.: R. Hiptmair.
Multigrid method for Maxwell's equations.
SIAM J. Numer. Anal., 36:204-225, 1999. MR 1654571 (99j:65229)
17.: P. Monk.
A posteriori error indicators for Maxwell's Equations.
J. Comp. Appl. Math., 100:173-190, 1998. MR 1659117 (2000k:78020)
18.: P. Monk,
Finite Element Methods for Maxwell's Equations.
Oxford University Press, 2003. MR 2059447 (2005d:65003)
19.: J.-C. Nedelec.
Mixed finite elements in ${\mathbb{R}}^3$ .
Numer. Math., 35:315-341, 1980. MR 592160 (81k:65125)
20.: J.-C. Nédélec.
A new family of mixed finite elements in ${\mathbb{R}}^3$ .
Numer. Math., 35:315-341, 1980. MR 864305 (88e:65145)
21.: S. Nicaise and E. Creusé.
A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes.
CALCOLO, 40:249-271, 2003. MR 2025712 (2004j:65133)
22.: J. Pasciak and J. Zhao.
Overlapping Schwarz methods in H(curl) on nonconvex domains.
East West J. Numer. Anal., 10:221-234, 2002. MR 1935967 (2003j:65108)
23.: P.-A. Raviart and J.-M. Thomas.
A mixed finite element method for second order elliptic problems.
In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, pages 292-315. Springer, Berlin, 1977. MR 0483555 (58:3547)
24.: L. R. Scott and S. Zhang.
Finite element interpolation of nonsmooth functions satisfying boundary conditions.
Math. Comp., 54(190):483-493, 1990. MR 1011446 (90j:65021)
25.: J. Schöberl.
Commuting quasi-interpolation operators for mixed finite elements.
Report ISC-01-10-MATH, Texas A&M University, available from www.isc.tamu.edu/iscpubs/iscreports.html, 2001.
26.: J. Schöberl and S. Zaglmayr.
High order Nédélec elements with local complete sequence properties.
Int. J. for Computation and Maths. in Electrical and Electronic Eng. COMPEL, to appear, 2005. MR 2169504
27.: J. Schöberl.
A multilevel decomposition result in $H(\operatorname{curl})$ .
Proceedings of the $8^{th}$ European Multigrid Conference EMG 2005, TU Delft (to appear).
28.: A. Toselli.
Overlapping Schwarz methods for Maxwell's equations in three dimension.
Numer. Math., 86:733-752, 2000. MR 1794350 (2001h:65137)
29.: A. Toselli and O. Widlund.
Domain Decomposition Methods - Algorithms and Theory.
Springer Series in Computational Mathematics, Vol. 34, 2005. MR 2104179 (2005g:65006)
30.: R. Verfürth.
A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques.
Wiley-Teubner, 1996.
31.: J. P. Webb,
Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,
IEEE Trans. on Antennas and Propagation, 47:1244-1253, 1999. MR 1711458 (2000g:78031)
32.: Sabine Zaglmayr,
High Order Finite Element Methods for Electromagnetic Field Computation. Ph.D. thesis, Institute for Computational Mathematics, Johannes Kepler University Linz, 2006

Joachim Schöberl
Affiliation: Center for Computational Engineering Science, RWTH Aachen University, Pauwelstrasse 19, D-52074 Aachen, Germany

DOI: 10.1090/S0025-5718-07-02030-3
PII: S 0025-5718(07)02030-3
Keywords: Cl\'ement operator, Maxwell equations, edge elements
Received by editor(s): May 5, 2005
Received by editor(s) in revised form: July 25, 2006
Posted: December 12, 2007
Additional Notes: The author acknowledges support from the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria, and from the Austrian Science Foundation FWF within project grant Start Y-192, ``hp-FEM: Fast Solvers and Adaptivity''
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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