Analysis of the spectral harmonically enriched multiscale coarse space (SHEM) in 2D

The Spectral Harmonically Enriched Multiscale (SHEM) coarse space for domain decomposition methods was introduced as a cheaper alternative to GenEO (Generalized Eigenvalue Problems in the Overlap) with similar performance for high contrast problems. In SHEM, one enriches the coarse space with specific, cheaply computable coarse space components to get faster convergence for domain decomposition methods. For high contrast problems, this enrichment leads to robustness against variations and discontinuities in the problem parameters both inside subdomains and across and along subdomain boundaries. We present and analyze here SHEM in 2D based on simple, sparse lower dimensional eigenvalue problems on the interfaces between subdomains, and also a variant that performs equally well in practice, and does not require the solve of eigenvalue problems at all. Our enrichment process naturally reaches the Optimal Harmonically Enriched Multiscale coarse space (OHEM) represented by the full discrete harmonic space. We give a complete convergence analysis of SHEM in 2D, and also test both SHEM variants and OHEM numerically in 2D.