Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations

Zhong, Liuqiang; Chen, Long; Shu, Shi; Wittum, Gabriel; Xu, Jinchao

doi:10.1090/S0025-5718-2011-02544-5

Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations
HTML articles powered by AMS MathViewer

by Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum and Jinchao Xu PDF

Math. Comp. 81 (2012), 623-642 Request permission

Abstract:

We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nédélec edge elements, for three-dimen- sional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is the establishment of a quasi-orthogonality and a localized a posteriori error estimator.

References

Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607, DOI 10.1090/S0025-5718-99-01013-3
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
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Rudolf Beck, Peter Deuflhard, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell’s equations, Surveys Math. Indust. 8 (1999), no. 3-4, 271–312. MR 1737416
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
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
C. Carstensen and R. H. W. Hoppe, Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations, J. Numer. Math. 13 (2005), no. 1, 19–32. MR 2130149, DOI 10.1163/1569395054069017
J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046, DOI 10.1137/07069047X
L. Chen, R. Nochetto, and J. Xu, Multilevel methods on graded bisection grids II: $H(curl)$ and $H(div)$ systems, Preprint, 2008.
Zhiming Chen, Long Wang, and Weiying Zheng, An adaptive multilevel method for time-harmonic Maxwell equations with singularities, SIAM J. Sci. Comput. 29 (2007), no. 1, 118–138. MR 2285885, DOI 10.1137/050636012
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, DOI 10.1090/S0025-5718-07-02081-9
Martin Costabel and Monique Dauge, Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal. 151 (2000), no. 3, 221–276. MR 1753704, DOI 10.1007/s002050050197
Martin Costabel, Monique Dauge, and Serge Nicaise, Singularities of eddy current problems, M2AN Math. Model. Numer. Anal. 37 (2003), no. 5, 807–831. MR 2020865, DOI 10.1051/m2an:2003056
D. Dauge and R. Stevenson, Sparse tensor product wavelet approximation of singular functions, SIAM J. Math. Anal. 42 (2010), no. 5, 2203–2228. Preprint 09-23, Universite de Rennes 1, (2009).
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Jayadeep Gopalakrishnan and Joseph E. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations, Math. Comp. 72 (2003), no. 241, 1–15. MR 1933811, DOI 10.1090/S0025-5718-01-01406-5
Jayadeep Gopalakrishnan, Joseph E. Pasciak, and Leszek F. Demkowicz, Analysis of a multigrid algorithm for time harmonic Maxwell equations, SIAM J. Numer. Anal. 42 (2004), no. 1, 90–108. MR 2051058, DOI 10.1137/S003614290139490X
R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
Ralf Hiptmair and Jinchao Xu, Nodal auxiliary space preconditioning in $\textbf {H}(\textbf {curl})$ and $\textbf {H}(\textrm {div})$ spaces, SIAM J. Numer. Anal. 45 (2007), no. 6, 2483–2509. MR 2361899, DOI 10.1137/060660588
Ralf Hiptmair and Weiying Zheng, Local multigrid in $\textbf {H}(\bf {curl})$, J. Comput. Math. 27 (2009), no. 5, 573–603. MR 2536903, DOI 10.4208/jcm.2009.27.5.012
R. H. W. Hoppe and J. Schöberl, Convergence of adaptive edge element methods for the 3D eddy currents equations, J. Comput. Math. 27 (2009), no. 5, 657–676. MR 2536907, DOI 10.4208/jcm.2009.27.5.016
F. Izsak and J. Van Der Vegt, A reliable and efficient implicit a posteriori error estimate technique for the time harmonic Maxwell equations, Tech. report, Internal report, Department of Applied Mathematics, University of Twente, Netherlands, 2007, http://eprints. eemcs. utwente. nl/11443/, M2AN, submitted for publication, 2007.
Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875, DOI 10.1016/0377-0427(94)90034-5
Khamron Mekchay and Ricardo H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), no. 5, 1803–1827. MR 2192319, DOI 10.1137/04060929X
William F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software 15 (1989), no. 4, 326–347 (1990). MR 1062496, DOI 10.1145/76909.76912
Peter Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), no. 2, 243–261. MR 1182977, DOI 10.1007/BF01385860
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
Peter Monk, A simple proof of convergence for an edge element discretization of Maxwell’s equations, Computational electromagnetics (Kiel, 2001) Lect. Notes Comput. Sci. Eng., vol. 28, Springer, Berlin, 2003, pp. 127–141. MR 1986135, DOI 10.1007/978-3-642-55745-3_{9}
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR 1980447, DOI 10.1137/S0036144502409093
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
Ricardo H. Nochetto, Kunibert G. Siebert, and Andreas Veeser, Theory of adaptive finite element methods: an introduction, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 409–542. MR 2648380, DOI 10.1007/978-3-642-03413-8_{1}2
J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements, 2nd European Conference on Computational Mechanics, 2001, pp. 854–855.
J. Schöberl, A multilevel decomposition result in $H(curl)$, Computing and Visualization in Science (2005), 41–52.
Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, DOI 10.1090/S0025-5718-07-02030-3
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
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Advances in Numerical Mathematics. Wiley-Teubner, Chichester-Stuttgart, 1996.
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Jinchao Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992), no. 2, 303–319. MR 1154268, DOI 10.1137/0729020
Jinchao Xu, Long Chen, and Ricardo H. Nochetto, Optimal multilevel methods for $H(\textrm {grad})$, $H(\textrm {curl})$, and $H(\textrm {div})$ systems on graded and unstructured grids, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659. MR 2648382, DOI 10.1007/978-3-642-03413-8_{1}4
L. Zhong, Fast algorithms of edge element discretizations and adaptive finite element methods for two classes of maxwell equations, Ph.D. thesis, Xiangtan University, 2009.
L. Zhong, S. Shu, J. Wang, and J. Xu, Two-grid methods and precondtioners for time-harmonic Maxwell equations, Numer. Linear Algebra Appl., 2011, accepted.
Liuqiang Zhong, Shi Shu, Gabriel Wittum, and Jinchao Xu, Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell’s equations, J. Comput. Math. 27 (2009), no. 5, 563–572. MR 2536902, DOI 10.4208/jcm.2009.27.5.011