Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

TL;DR

This paper develops robust domain decomposition preconditioners for high-contrast, multiscale elliptic PDEs on irregular domains, ensuring efficiency and stability independent of coefficient contrast.

Contribution

It introduces a new two-level additive Schwarz preconditioner using local spectral information for irregular domains with high-contrast coefficients.

Findings

Condition number bounded independently of contrast

Preconditioners effective on irregular domains

Numerical experiments confirm robustness

Abstract

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains. The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.