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Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation∗

Published online by Cambridge University Press:  31 July 2012

Huangxin Chen ,
Ronald H.W. Hoppe  and
Xuejun Xu
Huangxin Chen
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, P.R. China. chx@lsec.cc.ac.cn
Ronald H.W. Hoppe
Affiliation:
Department of Mathematics, University of Houston, Houston, 77204-3008 TX, USA; rohop@math.uh.edu Institute for Mathematics, University of Augsburg, 86159 Augsburg, Germany
Xuejun Xu
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, P.R. China; xxj@lsec.cc.ac.cn
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Abstract

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

Keywords

Maxwell equationsNédélec edge elementsindefinitemultigrid methodslocal Hiptmair smoothersadaptive edge finite element methodsoptimality
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 125 - 147
Copyright
© EDP Sciences, SMAI, 2012

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References

Aksoylu, B. and Holst, M., Optimality of multilevel preconditioners for local mesh refinement in three dimensions. SIAM J. Numer. Anal. 44 (2006) 1005–1025. Google Scholar
Aksoylu, B., Bond, S. and Holst, M., An odyssey into local refinement and multilevel preconditioning III : implementation and numerical experiments. SIAM J. Sci. Comput. 25 (2003) 478–498. Google Scholar
Arnold, D., Falk, R. and Winther, R., Multigrid in H(div) and H(curl). Numer. Math. 85 (2000) 197–218. Google Scholar
Bai, D. and Brandt, A., Local mesh refinement multilevel techniques. SIAM J. Sci. Stat. Comput. 8 (1987) 109–134. Google Scholar
Bänsch, E., Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Eng. 3 (1991) 181–191. Google Scholar
Beck, R., Deuflhard, P., Hiptmair, R., Hoppe, R.H.W. and Wohlmuth, B., Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Surv. Math. Indust. 8 (1999) 271–312. Google Scholar
Beck, R., Hiptmair, R., Hoppe, R.H.W. and Wohlmuth, B., Residual based a posteriori error estimators for eddy current computation. ESAIM : M2AN 34 (2000) 159–182. Google Scholar
A. Bossavit, Computational Electromagnetism : Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998).
J.H. Bramble, Multigrid Methods. Pitman (1993).
Bramble, J.H., Pasciak, J.E., Wang, J. and Xu, J., Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57 (1991) 23–45. Google Scholar
Bramble, J.H., Kwak, D.Y. and Pasciak, J.E., Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal. 31 (1994) 1746–1763. Google Scholar
Carstensen, C. and Hoppe, R.H.W., Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations. J. Numer. Math. 13 (2005) 19–32. Google Scholar
Chen, H. and Xu, X., Local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems. SIAM J. Numer. Anal. 47 (2010) 4492–4516. Google Scholar
Chen, Z., Wang, L. and Zheng, W., An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput. 29 (2007) 118–138. Google Scholar
Chen, J., Xu, Y. and Zou, J., Convergence analysis of an adaptive edge element method for Maxwell’s equations. Appl. Numer. Math. 59 (2009) 2950–2969. Google Scholar
Dahmen, W. and Kunoth, A., Multilevel preconditioning. Numer. Math. 63 (1992) 315–344. Google Scholar
Dörfler, W., A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Google Scholar
Gopalakrishnan, J. and Pasciak, J., Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations. Math. Comp. 72 (2003) 1–15. Google Scholar
Gopalakrishnan, J., Pasciak, J. and Demkowicz, L.F., Analysis of a multigrid algorithm for time harmonic Maxwell equations. SIAM J. Numer. Anal. 42 (2004) 90–108. Google Scholar
Hiptmair, R., Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36 (1998) 204–225. Google Scholar
Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. Google Scholar
Hiptmair, R. and Xu, J., Nodal auxiliary spaces preconditions in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483–2509. Google Scholar
Hiptmair, R. and Zheng, W., Local multigrid in H(curl,Ω). J. Comput. Math. 27 (2009) 573–603. Google Scholar
R. Hiptmair, H. Wu and W. Zheng, On uniform convergence theory of local multigrid methods in H 1(Ω) and H(curl,Ω). Preprint (2010).
Hoppe, R.H.W. and Schöberl, J., Convergence of adaptive edge element methods for the 3D eddy currents equations. J. Comput. Math. 27 (2009) 657–676. Google Scholar
R.H.W. Hoppe, X. Xu and H. Chen, Local Multigrid on Adaptively Refined Meshes and Multilevel Preconditioning with Applications to Problems in Electromagnetism and Acoustics, in Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations, edited by O. Axelsson and J. Karatson. Bentham, Bussum, The Netherlands (2010) 125–145.
Leis, R., Exterior boundary-value problems in mathematical physics, in Trends in Applications of Pure Mathematics to Mechanics, edited by H. Zorski. Monographs Stud. Math. 5 (1979) 187–203. Google Scholar
Monk, P., A posteriori error indicators for Maxwell’s equations. Comput. Appl. Math. 100 (1998) 173–190. Google Scholar
P. Monk, Finite element methods for Maxwell equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003).
Nédélec, J.-C., Mixed finite element in lR3. Numer. Math. 35 (1980) 315–341. Google Scholar
Nédélec, J.-C., A new family of mixed finite elements in lR3. Numer. Math. 50 (1986) 57–81. Google Scholar
P. Oswald, Multilevel Finite Element Approximation : Theory and Applications. Teubner, Stuttgart (1994).
Rüde, U., Fully adaptive multigrid methods. SIAM J. Numer. Anal. 30 (1993) 230–248. Google Scholar
Sterz, O., Hauser, A. and Wittum, G., Adaptive local multigrid methods for solving time-harmonic eddy current problems. IEEE Trans. Magn. 42 (2006) 309–318. Google Scholar
L. Tartar, Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin, Heidelberg, New York (2007).
H. Whitney, Geometric Integration Theory. Princeton University Press, Princeton (1957).
Wu, H.J. and Chen, Z.M., Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China 39 (2006) 1405–1429. Google Scholar
J. Xu, L. Chen and R. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation. Springer (2009) 599–659.
Xu, X., Chen, H. and Hoppe, R.H.W., Optimality of local multilevel methods on adaptively refined meshes for elliptic boundary value problems. J. Numer. Math. 18 (2010) 59–90. Google Scholar
X. Xu, H. Chen and R.H.W. Hoppe, Optimality of local multilevel methods for adaptive nonconforming P1 finite element methods. J. Comput. Math. (2012), in press.
Zhong, L., Chen, L. and Xu, J., Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numer. Lin. Algebra Appl. 17 (2009) 415–432. Google Scholar
Zhong, L., Chen, L., Shu, S., Wittum, G. and Xu, J., Quasi-optimal convergence of adaptive edge finite element methods for three dimensional indefinite time-harmonic Maxwell’s equations. Math. Comp. 81 (2012), 623–642. Google Scholar