Spectral analysis of the anisotropic Steklov–Poincaré matrix

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.