Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems

TL;DR

This paper introduces an enhanced additive average Schwarz preconditioner with adaptive coarse spaces, achieving scalable and robust solutions for multiscale elliptic problems with highly variable coefficients.

Contribution

It proposes two new adaptive coarse spaces that improve the additive average Schwarz method, maintaining optimal scalability and robustness against coefficient discontinuities.

Findings

Condition number is independent of coefficient variations.

Preconditioner retains optimal scalability with mesh ratio H/h.

Method is effective for multiscale elliptic problems.

Abstract

We present an analysis of the additive average Schwarz preconditioner with two newly proposed adaptively enriched coarse spaces which was presented at the 23rd International conference on domain decomposition methods in Korea, for solving second order elliptic problems with highly varying and discontinuous coefficients. It is shown that the condition number of the preconditioned system is bounded independently of the variations and the jumps in the coefficient, and depends linearly on the mesh parameter ratio H/h, that is the ratio between the subdomain size and the mesh size, thereby retaining the same optimality and scalablity of the original additive average Schwarz preconditioner.