A posteriori error estimates for Maxwell equations

Schöberl, Joachim

doi:10.1090/S0025-5718-07-02030-3

A posteriori error estimates for Maxwell equations
HTML articles powered by AMS MathViewer

by Joachim Schöberl;

Math. Comp. 77 (2008), 633-649

DOI: https://doi.org/10.1090/S0025-5718-07-02030-3

Published electronically: December 12, 2007

PDF | Request permission

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.

References

Mark Ainsworth and Joe Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes, Internat. J. Numer. Methods Engrg. 58 (2003), no. 14, 2103–2130. MR 2022172, DOI 10.1002/nme.847
Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
Rudi Beck, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159–182 (English, with English and French summaries). MR 1735971, DOI 10.1051/m2an:2000136
R. Beck, R. Hiptmair, and B. Wohlmuth, Hierarchical error estimator for eddy current computation, Numerical mathematics and advanced applications (Jyväskylä, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 110–120. MR 1936173
Daniele Boffi, Fortin operator and discrete compactness for edge elements, Numer. Math. 87 (2000), no. 2, 229–246. MR 1804657, DOI 10.1007/s002110000182
D. Boffi, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett. 14 (2001), no. 1, 33–38. MR 1793699, DOI 10.1016/S0893-9659(00)00108-7
A. Bossavit, Mixed finite elements and the complex of Whitney forms, The mathematics of finite elements and applications, VI (Uxbridge, 1987) Academic Press, London, 1988, pp. 137–144. MR 956893
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.
Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. no. , no. R-2, 77–84 (English, with French summary). MR 400739
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.
L. Demkowicz and I. Babuška. Optimal $p$ interpolation error estimates for edge finite elements of variable order in $2d$. Technical Report 01-11, TICAM, University of Texas at Austin, 2001.
L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz, de Rham diagram for $hp$ finite element spaces, Comput. Math. Appl. 39 (2000), no. 7-8, 29–38. MR 1746160, DOI 10.1016/S0898-1221(00)00062-6
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
Peter Monk, A posteriori error indicators for Maxwell’s equations, J. Comput. Appl. Math. 100 (1998), no. 2, 173–190. MR 1659117, DOI 10.1016/S0377-0427(98)00187-3
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
S. Nicaise and E. Creusé, A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes, Calcolo 40 (2003), no. 4, 249–271. MR 2025712, DOI 10.1007/s10092-003-0077-y
J. E. Pasciak and J. Zhao, Overlapping Schwarz methods in $H$(curl) on polyhedral domains, J. Numer. Math. 10 (2002), no. 3, 221–234. MR 1935967, DOI 10.1515/JNMA.2002.221
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 292–315. MR 483555
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
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.
Joachim Schöberl and Sabine Zaglmayr, High order Nédélec elements with local complete sequence properties, COMPEL 24 (2005), no. 2, 374–384. MR 2169504, DOI 10.1108/03321640510586015
J. Schöberl. A multilevel decomposition result in $H(\operatorname {curl})$. Proceedings of the $8^{th}$ European Multigrid Conference EMG 2005, TU Delft (to appear).
Andrea Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions, Numer. Math. 86 (2000), no. 4, 733–752. MR 1794350, DOI 10.1007/PL00005417
Andrea Toselli and Olof Widlund, Domain decomposition methods—algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR 2104179, DOI 10.1007/b137868
R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996.
Jon P. Webb, Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements, IEEE Trans. Antennas and Propagation 47 (1999), no. 8, 1244–1253. MR 1711458, DOI 10.1109/8.791939
Sabine Zaglmayr, High Order Finite Element Methods for Electromagnetic Field Computation. Ph.D. thesis, Institute for Computational Mathematics, Johannes Kepler University Linz, 2006