Low-rank approximation of elliptic boundary value problems with high-contrast coefficients

TL;DR

This paper demonstrates that hierarchical matrix approximations for elliptic boundary value problems with high-contrast coefficients converge independently of the contrast, enabling efficient, coefficient-agnostic numerical methods.

Contribution

The paper proves contrast-independent convergence of degenerate Green's function approximations, facilitating fast hierarchical matrix methods for high-contrast elliptic problems.

Findings

Convergence is independent of coefficient contrast

Hierarchical matrix approximations are effective without coefficient adaptation

Provides theoretical foundation for contrast-robust numerical methods

Abstract

We analyze the convergence of degenerate approximations to Green's function of elliptic boundary value problems with high-contrast coefficients. It is shown that the convergence is independent of the contrast if the error is measured with respect to suitable norms. This lays ground to fast methods (so-called hierarchical matrix approximations) which do not have to be adapted to the coefficients.