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 (2012)

  • 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 47.1 (2012): 125-147. <http://eudml.org/doc/222151>.

@article{Chen2012,
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},
keywords = {Maxwell equations; Nédélec edge elements; indefinite; multigrid methods; local Hiptmair smoothers; adaptive edge finite element methods; optimality; adaptive mesh refinement; a posteriori error estimators; multilevel iterative schemes; convergence; numerical examples},
language = {eng},
month = {7},
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/222151},
volume = {47},
year = {2012},
}

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
DA - 2012/7//
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; adaptive mesh refinement; a posteriori error estimators; multilevel iterative schemes; convergence; numerical examples
UR - http://eudml.org/doc/222151
ER -

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