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

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

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

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topChen, 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 -

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