The anisotropic Steklov-Poincare matrix
Résumé
In this work we analyse the Steklov-Poincare (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(hé2). 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.