research-article

Multigrid Preconditioning for Krylov Methods for Time-Harmonic Maxwell's Equations in Three Dimensions

Authors:

U. M. AscherAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 24, Issue 2

Pages 702 - 718

https://doi.org/10.1137/S1064827501387358

Published: 01 January 2002 Publication History

Abstract

We consider the rapid simulation of three-dimensional electromagnetic problems in geophysical parameter regimes, where the conductivity may vary significantly and the range of frequencies is moderate. Toward developing a multigrid preconditioner, we present a Fourier analysis based on a finite-volume discretization of a vector potential formulation of time-harmonic Maxwell's equations on a staggered grid in three dimensions. We prove grid-independent bounds on the eigenvalue and singular value ranges of the system obtained using a preconditioner based on exact inversion of the dominant diagonal blocks of the non-Hermitian coefficient matrix. This result implies that a preconditioner that uses single multigrid cycles to effect inversion of the diagonal blocks also yields a preconditioned system with an $\ell_2$-condition number bounded independent of the grid size.

We then present numerical examples for more realistic situations involving large variations in conductivity (i.e.,\ jump discontinuities). Block-preconditioning with one multigrid cycle using Dendy's BOXMG solver is found to yield convergence in very few iterations, apparently independent of the grid size. The experiments show that the somewhat restrictive assumptions of the Fourier analysis do not prohibit it from describing the essential local behavior of the preconditioned operator under consideration. A very efficient, practical solver is obtained.

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Cited By

Mulder W(2008)Geophysical modelling of 3D electromagnetic diffusion with multigridComputing and Visualization in Science10.5555/3113216.311351811:3(129-138)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.5555/3113216.3113518
Baranoski GWan JRokne JBell I(2005)Simulating the dynamics of auroral phenomenaACM Transactions on Graphics10.1145/1037957.103796024:1(37-59)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1145/1037957.1037960

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Multigrid Preconditioning for Krylov Methods for Time-Harmonic Maxwell's Equations in Three Dimensions

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 24, Issue 2

2002

360 pages

ISSN:1064-8275

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Copyright © 2002 Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2002

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Cited By

Mulder W(2008)Geophysical modelling of 3D electromagnetic diffusion with multigridComputing and Visualization in Science10.5555/3113216.311351811:3(129-138)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.5555/3113216.3113518
Baranoski GWan JRokne JBell I(2005)Simulating the dynamics of auroral phenomenaACM Transactions on Graphics10.1145/1037957.103796024:1(37-59)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1145/1037957.1037960

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