Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-30T17:46:45.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

Published online by Cambridge University Press:  28 May 2015

Ralf Hiptmair ,
Haijun Wu  and
Weiying Zheng
Ralf Hiptmair*
Affiliation:
SAM, ETH Zürich, CH-8092 Zürich, Swizerland
Haijun Wu*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, China
Weiying Zheng*
Affiliation:
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
*
Corresponding author.Email address:hiptmair@sam.math.ethz.ch
Corresponding author.Email address:hjw@nju.edu.cn
Corresponding author.Email address:zwy@lsec.cc.ac.en
Get access

Abstract

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell’s equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their “immediate” neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.

Keywords

65N3065N5578A25MMaxwell’s equationsLagrangian finite elementsedge elementsadaptive multigrid methodsuccessive subspace correction
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 3 , August 2012 , pp. 297 - 332
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, D., Falk, R., and Winther, R., Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), pp. 175–195.CrossRefGoogle Scholar
[2]Arnold, D., Falk, R., and Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15 (2006), pp. 1–155.Google Scholar
[3]Bai, D. and Brandt, A., Local mesh refinement multilevel techniques, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 109–134.CrossRefGoogle Scholar
[4]Beck, R., Deuflhard, P., Hiptmair, R., Hoppe, R., and Wohlmuth, B., Adaptive multilevel methods for edge element discretizations of Maxwell’s equations, Surveys on Mathematics for Industry, 8 (1998), pp. 271–312.Google Scholar
[5]Beck, R., Hiptmair, R., Hoppe, R., and Wohlmuth, B., Residual based a-posteriori error estimators for eddy current computation, M2AN, 34 (2000), pp. 159–182.CrossRefGoogle Scholar
[6]Binev, P., Dahmen, W., and Devore, R., Adaptive finite element methods with convergence rates, Numerische Mathematik, 97 (2004), pp. 219–268.CrossRefGoogle Scholar
[7]Bornemann, F. and Yserentant, H., A basic norm equivalence for the theory of multilevel methods, Numer. Math., 64 (1993), pp. 455–476.CrossRefGoogle Scholar
[8]Bramble, J., Multigrid Methods, vol. 294 of Pitman Research Notes in Mathematical Sciences, Longman, Essex, 1993.Google Scholar
[9]Cascon, J. M., Kreuzer, C., Nochetto, R. H., and Siebert, K. G., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.CrossRefGoogle Scholar
[10]Chen, Z. and Dai, S., On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput., 24 (2002), p. 443ÍC462.CrossRefGoogle Scholar
[11]Chen, Z., Wang, L., and Zheng, W., An adaptive multilevel method for time-harmonic maxwell equations with singularities, SIAM J. Sci. Comp., 29 (2007), pp. 118–138.CrossRefGoogle Scholar
[12]Clement, P., Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 2 (1975), pp. 77–84.Google Scholar
[13]Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.CrossRefGoogle Scholar
[14]Hiptmair, R., Multigrid method for Maxwell’s equations, Tech. Report 374, Institut für Mathematik, Universität Augsburg, 1997. USE HIP 99.Google Scholar
[15]Hiptmair, R., Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal., 36 (1999), pp. 204–225.Google Scholar
[16]Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), pp. 237–339.CrossRefGoogle Scholar
[17]Hiptmair, R., Analysis of multilevel methods for eddy current problems, Math. Comp., 72 (2003), pp. 1281–1303.Google Scholar
[18]Hiptmair, R. and Xu, J., Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509.CrossRefGoogle Scholar
[19]Hiptmair, R. and Zheng, W., Local multigrid in H(curl), J. Comp. Math., 27 (2009), pp. 573–603.Google Scholar
[20]Kossaczký, I., A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math., 55 (1994), pp. 275–288.CrossRefGoogle Scholar
[21]Maubach, J., Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Stat. Comp., 16 (1995), pp. 210–227.CrossRefGoogle Scholar
[22]Mitchell, W., Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Stat. Comput, 13 (1992), pp. 146–167.CrossRefGoogle Scholar
[23]Oswald, P., Multilevel finite element approximation, Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart, 1994.Google Scholar
[24]Schmidt, A. and Siebert, K., Alberta-an adaptive hierarchical finite element toolbox, tech. report, ALBERTA is available online from: http://www.alberta-fem.de.Google Scholar
[25]Schöberl, J., Commuting quasi-interpolation operators for mixed finite elements, Preprint ISC-01-10-MATH, Texas A&M University, College Station, TX, 2001.Google Scholar
[26]Schöberl, J., A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), pp. 633–649.Google Scholar
[27]Scott, L. R. and Zhang, Z., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[28]Sterz, O., Hauser, A., and Wittum, G., Adaptive local multigrid methods for solving time-harmonic eddy current problems, IEEE Trans. Magnetics, 42 (2006), pp. 309–318.CrossRefGoogle Scholar
[29]Stevenson, R., Optimality of a standard adaptive finite element method, Foundations of Computational Mathematics, 7 (2007), pp. 245–269.CrossRefGoogle Scholar
[30]Toselli, A. and Widlund, O., Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, Berlin Heiderberg, 2005.Google Scholar
[31]Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement rechniques, Wiley, Chichester, Stuttgart, 1996.Google Scholar
[32]Wu, H.-J. and Chen, Z.-M., Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems, Science in China: Series A Mathematics, 49 (2006), pp. 1–28.CrossRefGoogle Scholar
[33]Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992), pp. 581–613.CrossRefGoogle Scholar
[34]Xu, J., An introduction to multilevel methods, in Wavelets, Multilevel Methods and Elliptic PDEs, Ainsworth, M., Levesley, K., Marletta,, M. and Light, W, eds., Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 1997, pp. 213–301.Google Scholar
[35]Xu, J., Chen, L., and Nochetto, R., Optimal multilevel methods for H(grad), H (curl), H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation, DeVore, R. and Kunoth, A., eds., Springers, 2009, pp. 599–659.CrossRefGoogle Scholar
[36]Xu, J. and Zhu, Y.-R., Uniformly convergent multigrid methods for elliptic problems with strongly discontinuous coefficients, Math. Models Methods Appl. Sci., 18 (2008), pp. 77–105.CrossRefGoogle Scholar
[37]Xu, J. and Zikatanov, L., The method of alternating projections and the method of subspace corrections in Hilbert space, J. Am. Math. Soc, 15 (2002), pp. 573–597.CrossRefGoogle Scholar
[38]Yserentant, H., On the multi-level splitting of finite element spaces, Numer. Math., 58 (1986), pp. 379–412.CrossRefGoogle Scholar
[39]Zheng, Z., Chen, Z., and Wang, L., An adaptive finite element method for the h–ψ formulation of time-dependent eddy current problems, Numer. Math., 103 (2006), pp. 667–689.CrossRefGoogle Scholar