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

Huangxin Chen; Ronald H. W. Hoppe; Xuejun Xu

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 125-147
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Chen, Huangxin, Hoppe, Ronald H. W., and Xu, Xuejun. "Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 125-147. <http://eudml.org/doc/273310>.

@article{Chen2013,
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.},
author = {Chen, Huangxin, Hoppe, Ronald H. W., Xu, Xuejun},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Maxwell equations; nédélec edge elements; indefinite; multigrid methods; local hiptmair smoothers; adaptive edge finite element methods; optimality; Nédélec edge elements; local Hiptmair smoothers; adaptive mesh refinement; a posteriori error estimators; multilevel iterative schemes; convergence; numerical examples},
language = {eng},
number = {1},
pages = {125-147},
publisher = {EDP-Sciences},
title = {Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation},
url = {http://eudml.org/doc/273310},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Chen, Huangxin
AU - Hoppe, Ronald H. W.
AU - Xu, Xuejun
TI - Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 125
EP - 147
AB - 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.
LA - eng
KW - Maxwell equations; nédélec edge elements; indefinite; multigrid methods; local hiptmair smoothers; adaptive edge finite element methods; optimality; Nédélec edge elements; local Hiptmair smoothers; adaptive mesh refinement; a posteriori error estimators; multilevel iterative schemes; convergence; numerical examples
UR - http://eudml.org/doc/273310
ER -

References

top
  1. [1] B. Aksoylu and M. Holst, Optimality of multilevel preconditioners for local mesh refinement in three dimensions. SIAM J. Numer. Anal.44 (2006) 1005–1025. Zbl1153.65093MR2231853
  2. [2] B. Aksoylu, S. Bond and M. Holst, An odyssey into local refinement and multilevel preconditioning III : implementation and numerical experiments. SIAM J. Sci. Comput.25 (2003) 478–498. Zbl1048.65104MR2058071
  3. [3] D. Arnold, R. Falk and R. Winther, Multigrid in H(div) and H(curl). Numer. Math.85 (2000) 197–218. Zbl0974.65113MR1754719
  4. [4] D. Bai and A. Brandt, Local mesh refinement multilevel techniques. SIAM J. Sci. Stat. Comput.8 (1987) 109–134. Zbl0619.65091MR879406
  5. [5] E. Bänsch, Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Eng.3 (1991) 181–191. Zbl0744.65074MR1141298
  6. [6] R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Surv. Math. Indust.8 (1999) 271–312. Zbl0939.65136MR1737416
  7. [7] R. Beck, R. Hiptmair, R.H.W. Hoppe and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation. ESAIM : M2AN 34 (2000) 159–182. Zbl0949.65113MR1735971
  8. [8] A. Bossavit, Computational Electromagnetism : Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998). Zbl0945.78001MR1488417
  9. [9] J.H. Bramble, Multigrid Methods. Pitman (1993). Zbl0786.65094MR1247694
  10. [10] J.H. Bramble, J.E. Pasciak, J. Wang and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp.57 (1991) 23–45. Zbl0754.65085MR1090464
  11. [11] J.H. Bramble, D.Y. Kwak and J.E. Pasciak, Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal.31 (1994) 1746–1763. Zbl0813.65130MR1302683
  12. [12] 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) 19–32. Zbl1073.78008MR2130149
  13. [13] H. Chen and X. Xu, Local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems. SIAM J. Numer. Anal.47 (2010) 4492–4516. Zbl1209.65132MR2595046
  14. [14] Z. Chen, L. Wang and W. Zheng, An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput.29 (2007) 118–138. Zbl1136.78013MR2285885
  15. [15] J. Chen, Y. Xu and J. Zou, Convergence analysis of an adaptive edge element method for Maxwell’s equations. Appl. Numer. Math.59 (2009) 2950–2969. Zbl1183.78032MR2560827
  16. [16] W. Dahmen and A. Kunoth, Multilevel preconditioning. Numer. Math.63 (1992) 315–344. Zbl0757.65031MR1186345
  17. [17] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.33 (1996) 1106–1124. Zbl0854.65090MR1393904
  18. [18] J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations. Math. Comp.72 (2003) 1–15. Zbl1009.78009MR1933811
  19. [19] J. Gopalakrishnan, J. Pasciak and L.F. Demkowicz, Analysis of a multigrid algorithm for time harmonic Maxwell equations. SIAM J. Numer. Anal.42 (2004) 90–108. Zbl1079.78025MR2051058
  20. [20] R. Hiptmair, Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal.36 (1998) 204–225. Zbl0922.65081MR1654571
  21. [21] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer.11 (2002) 237–339. Zbl1123.78320MR2009375
  22. [22] R. Hiptmair and J. Xu, Nodal auxiliary spaces preconditions in H(curl) and H(div) spaces. SIAM J. Numer. Anal.45 (2007) 2483–2509. Zbl1153.78006MR2361899
  23. [23] R. Hiptmair and W. Zheng, Local multigrid in H(curl,Ω). J. Comput. Math.27 (2009) 573–603. Zbl1212.65486MR2536903
  24. [24] R. Hiptmair, H. Wu and W. Zheng, On uniform convergence theory of local multigrid methods in H1(Ω) and H(curl,Ω). Preprint (2010). 
  25. [25] R.H.W. Hoppe and J. Schöberl, Convergence of adaptive edge element methods for the 3D eddy currents equations. J. Comput. Math.27 (2009) 657–676. Zbl1212.65126MR2536907
  26. [26] 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. 
  27. [27] R. Leis, 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. Zbl0414.73082MR566529
  28. [28] P. Monk, A posteriori error indicators for Maxwell’s equations. Comput. Appl. Math.100 (1998) 173–190. Zbl1023.78004MR1659117
  29. [29] P. Monk, Finite element methods for Maxwell equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003). Zbl1024.78009MR2059447
  30. [30] J.-C. Nédélec, Mixed finite element in lR3. Numer. Math.35 (1980) 315–341. Zbl0419.65069
  31. [31] J.-C. Nédélec, A new family of mixed finite elements in lR3. Numer. Math.50 (1986) 57–81. Zbl0625.65107MR864305
  32. [32] P. Oswald, Multilevel Finite Element Approximation : Theory and Applications. Teubner, Stuttgart (1994). Zbl0830.65107MR1312165
  33. [33] U. Rüde, Fully adaptive multigrid methods. SIAM J. Numer. Anal.30 (1993) 230–248. Zbl0849.65090MR1202664
  34. [34] O. Sterz, A. Hauser and G. Wittum, Adaptive local multigrid methods for solving time-harmonic eddy current problems. IEEE Trans. Magn.42 (2006) 309–318. 
  35. [35] L. Tartar, Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin, Heidelberg, New York (2007). Zbl1126.46001MR2328004
  36. [36] H. Whitney, Geometric Integration Theory. Princeton University Press, Princeton (1957). Zbl0083.28204MR87148
  37. [37] H.J. Wu and Z.M. Chen, Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China39 (2006) 1405–1429. Zbl1112.65104MR2287269
  38. [38] 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. Zbl1193.65209MR2648382
  39. [39] X. Xu, H. Chen and R.H.W. Hoppe, Optimality of local multilevel methods on adaptively refined meshes for elliptic boundary value problems. J. Numer. Math.18 (2010) 59–90. Zbl1194.65147MR2629823
  40. [40] 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. Zbl1289.65274
  41. [41] L. Zhong, L. Chen and J. Xu, Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numer. Lin. Algebra Appl.17 (2009) 415–432. Zbl1240.65338MR2650219
  42. [42] L. Zhong, L. Chen, S. Shu, G. Wittum and J. Xu, Quasi-optimal convergence of adaptive edge finite element methods for three dimensional indefinite time-harmonic Maxwell’s equations. Math. Comp.81 (2012), 623–642. Zbl1263.78012MR2869030

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