Volume 5, Issue 3
Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

Ralf Hiptmair, Haijun Wu & Weiying Zheng

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 297-332.

Published online: 2012-05

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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

MMaxwell's equations, Lagrangian finite elements, edge elements, adaptive multigrid method, successive subspace correction.

  • AMS Subject Headings

65N30, 65N55, 78A25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-297, author = {}, title = {Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {297--332}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1128}, url = {http://global-sci.org/intro/article_detail/nmtma/5940.html} }
TY - JOUR T1 - Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 297 EP - 332 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1128 UR - https://global-sci.org/intro/article_detail/nmtma/5940.html KW - MMaxwell's equations, Lagrangian finite elements, edge elements, adaptive multigrid method, successive subspace correction. AB -

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.

Ralf Hiptmair, Haijun Wu & Weiying Zheng. (2020). Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 297-332. doi:10.4208/nmtma.2012.m1128
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