In this work we analyse the Steklov–Poincaré (or interface Schur complement) matrix arising in a domain decomposition method in the presence of anisotropy. Our problem is formu- lated such that three types of anisotropy are being considered: refinements with high aspect ratios, uniform refinements of a domain with high aspect ratio and anisotropic diffusion prob- lems discretized on uniform meshes. Our analysis indicates a condition number of the interface Schur complement with an order ranging from O(1) to O(h−2). By relating this behaviour to an underlying scale of fractional Sobolev spaces, we pro- pose optimal preconditioners which are spectrally equivalent to fractional matrix powers of a discrete interface Laplacian. Numerical experiments to validate the analysis are included; extensions to general domains and non-uniform meshes are also considered.

Spectral analysis of the anisotropic Steklov-Poincaré matrix

Arioli M;
2015

Abstract

In this work we analyse the Steklov–Poincaré (or interface Schur complement) matrix arising in a domain decomposition method in the presence of anisotropy. Our problem is formu- lated such that three types of anisotropy are being considered: refinements with high aspect ratios, uniform refinements of a domain with high aspect ratio and anisotropic diffusion prob- lems discretized on uniform meshes. Our analysis indicates a condition number of the interface Schur complement with an order ranging from O(1) to O(h−2). By relating this behaviour to an underlying scale of fractional Sobolev spaces, we pro- pose optimal preconditioners which are spectrally equivalent to fractional matrix powers of a discrete interface Laplacian. Numerical experiments to validate the analysis are included; extensions to general domains and non-uniform meshes are also considered.
Domain decomposition
Steklov–Poincaré operator
Anisotropic operators
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12572/6188
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