On overlapping domain decomposition methods for high-contrast multiscale problems

TL;DR

This paper reviews and proposes advanced overlapping domain decomposition methods for high-contrast multiscale problems, focusing on spectral-based coarse spaces that improve convergence and reduce iteration counts.

Contribution

It introduces novel spectral-based coarse space constructions that enhance the performance of domain decomposition algorithms for multiscale problems with high contrast.

Findings

Condition number independent of contrast and small scales

Near-optimal iteration counts with minimal coarse spaces

Effective localization of global fields improves convergence

Abstract

We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems and propose a domain decomposition method better performance in terms of the number of iterations. The main novelty of our approaches is the construction of coarse spaces, which are computed using spectral information of local bilinear forms. We present several approaches to incorporate the spectral information into the coarse problem in order to obtain minimal coarse space dimension. We show that using these coarse spaces, we can obtain a domain decomposition preconditioner with the condition number independent of contrast and small scales. To minimize further the number of iterations until convergence, we use this minimal dimensional coarse spaces in a construction combining them with large overlap local problems that take advantage of the possibility of localizing global fields orthogonal to the coarse space. We obtain a condition number close to 1 for the new method. We discuss possible drawbacks and further extensions.