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

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