Analysis of a New Harmonically Enriched Multiscale Coarse Space for Domain Decomposition Methods

TL;DR

This paper introduces a novel harmonically enriched multiscale coarse space for domain decomposition, enhancing robustness against parameter variations and enabling direct solving capabilities, validated through extensive numerical testing.

Contribution

It develops spectral and non-spectral variants of a harmonically enriched coarse space, improving robustness and efficiency in domain decomposition methods for high contrast problems.

Findings

The spectral method achieves robustness through eigenvalue problem solutions.

The non-spectral method offers practical efficiency without eigenvalue computations.

Numerical tests confirm the theoretical robustness and effectiveness of the proposed methods.

Abstract

We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition methods. For a coercive high contrast model problem, we show how to enrich the coarse space so that the method is robust against any variations and discontinuities in the problem parameters both inside subdomains and across and along subdomain boundaries. We prove our results for an enrichment strategy based on solving simple, lower dimensional eigenvalue problems on the interfaces between subdomains, and we call the resulting coarse space the spectral harmonically enriched multiscale coarse space (SHEM). We then also give a variant that performs equally well in practice, and does not require the solve of eigenvalue problems, which we call non-spectral harmonically enriched multiscale coarse space (NSHEM). Our enrichment process naturally reaches the optimal coarse space represented by the full discrete harmonic space, which enables us to turn the method into a direct solver (OHEM). We also extensively test our new coarse spaces numerically, and the results confirm our analysis