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Local fourier analysis of multigrid for the curl-curl equation

Boonen, Tim; Van Lent, Jan; Vandewalle, Stefan

Authors

Tim Boonen

Stefan Vandewalle



Abstract

We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk, and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. This procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows several well known properties to be recovered in Fourier space, such as the commutation properties of the classical Nédélec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to twoand three-dimensional problems in H(curl) and H(div) and to other types of regular meshes. © 2008 Society for Industrial and Applied Mathematics.

Citation

Boonen, T., Van Lent, J., & Vandewalle, S. (2007). Local fourier analysis of multigrid for the curl-curl equation. SIAM Journal on Scientific Computing, 30(4), 1730-1755. https://doi.org/10.1137/070679119

Journal Article Type Article
Publication Date Dec 1, 2007
Journal SIAM Journal on Scientific Computing
Print ISSN 1064-8275
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 30
Issue 4
Pages 1730-1755
DOI https://doi.org/10.1137/070679119
Keywords multigrid, curl-curl equation, local Fourier analysis
Public URL https://uwe-repository.worktribe.com/output/1013278
Publisher URL http://dx.doi.org/10.1137/070679119