A GenEO-type coarse space with smaller eigenproblems
Peter Bastian, Nils Friess
TL;DR
The paper tackles the setup cost of adaptive GenEO coarse spaces in two-level Schwarz methods for PDEs with highly heterogeneous coefficients. It introduces the R-GenEO coarse space, which solves local eigenproblems on a boundary-strip $\Omega_j^*$ and uses energy-minimizing extensions to generate coarse basis functions, preserving a coefficient-robust condition-number bound similar to GenEO. The authors prove a robustness bound $\kappa$ that mirrors the GenEO bound, with a potential trade-off in coarse-space size. Numerical experiments on a high-contrast elliptic problem demonstrate a roughly threefold reduction in setup time while maintaining comparable convergence, highlighting practical improvements for large-scale parallel solvers.
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.