A GenEO-type coarse space with smaller eigenproblems

TL;DR

This paper introduces a new variant of the GenEO coarse space that reduces setup costs by solving smaller eigenproblems near subdomain boundaries, maintaining robustness in overlapping Schwarz methods.

Contribution

A novel GenEO-type coarse space that involves solving eigenproblems only in boundary strips, reducing setup costs while preserving robustness.

Findings

Significant reduction in eigenproblem size and setup cost.

Maintains coefficient-robust condition number estimates.

Potentially larger coarse space for similar robustness.

Abstract

Coarse spaces are essential to ensure robustness w.r.t. the number of subdomains in two-level overlapping Schwarz methods. Robustness with respect to the coefficients of the underlying partial differential equation (PDE) can be achieved by adaptive (or spectral) coarse spaces involving the solution of local eigenproblems. The solution of these eigenproblems, although scalable, entails a large setup cost which may exceed the cost for the iteration phase. In this paper we present and analyse a new variant of the GenEO (Generalised Eigenproblems in the Overlap) coarse space which involves solving eigenproblems only in a strip connected to the boundary of the subdomain. This leads to a significant reduction of the setup cost while the method satisfies a similar coefficient-robust condition number estimate as the original method, albeit with a possibly larger coarse space.