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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.
@article{M2AN_2013__47_1_125_0,
author = {Chen, Huangxin and Hoppe, Ronald H. W. and Xu, Xuejun},
title = {Uniform convergence of local multigrid methods for the time-harmonic {Maxwell} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {125--147},
publisher = {EDP-Sciences},
volume = {47},
number = {1},
year = {2013},
doi = {10.1051/m2an/2012023},
mrnumber = {2968698},
zbl = {1278.65167},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012023/}
}
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 PY - 2013 SP - 125 EP - 147 VL - 47 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012023/ DO - 10.1051/m2an/2012023 LA - en ID - M2AN_2013__47_1_125_0 ER -
%0 Journal Article %A Chen, Huangxin %A Hoppe, Ronald H. W. %A Xu, Xuejun %T Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 125-147 %V 47 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012023/ %R 10.1051/m2an/2012023 %G en %F M2AN_2013__47_1_125_0
Chen, Huangxin; Hoppe, Ronald H. W.; Xu, Xuejun. Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 125-147. doi: 10.1051/m2an/2012023
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