Uniform Convergence of Local Multigrid Methods for the Time-harmonic Maxwell Equation

  • 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 Nedelec'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 isFor 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 Nedelec'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.show moreshow less

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Metadaten
Author:Huangxin Chen, Ronald H. W. HoppeGND, Xuejun Xu
URN:urn:nbn:de:bvb:384-opus4-11943
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/1472
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2011-01)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:Chinese Academy of Sciences, University of Houston
Release Date:2011/01/04
Tag:time-harmonic Maxwell equation; local multigrid methods; Hiptmair smoothers; adaptive edge element methods; optimality
GND-Keyword:Maxwellsche Gleichungen; Mehrgitterverfahren; Numerisches Verfahren
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand