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 formulated 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 problems 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 propose 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.
Original language | English |
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Pages (from-to) | 168-183 |
Number of pages | 16 |
Journal | Linear Algebra and its Applications |
Volume | 488 |
Early online date | 2 Oct 2015 |
DOIs | |
Publication status | Published - 1 Jan 2016 |
Keywords
- Domain decomposition
- Steklov–Poincaré operator
- Anisotropic operators
- Spectral analysis