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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
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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

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