Mathematical Analysis of Robustness of Two-Level Domain Decomposition Methods with respect to Inexact Coarse Solves

TL;DR

This paper analyzes how the robustness of GenEO-based two-level domain decomposition methods is affected by inexact coarse solves, proposing modifications to ensure their effectiveness in large-scale problems.

Contribution

It provides a mathematical analysis of the robustness of GenEO methods with inexact coarse solves and introduces modifications to GenEO-2 for maintaining robustness.

Findings

GenEO methods are robust with exact coarse solves.

Inexact coarse solves can affect the robustness of the methods.

Modified GenEO-2 maintains robustness with inexact solves.

Abstract

Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. The GenEO coarse space has been shown to lead to a fully robust two-level Schwarz preconditioner which scales well over multiple cores [27, 19] as has been proved rigorously in [27]. The robustness is due to its good approximation properties for problems with highly heterogeneous material parameters. It is available in the finite element packages FreeFem++, Feel++ and recently in Dune and is implemented as a standalone library in HPDDM But the coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large and exact solves are used. It is therefore interesting to consider the effect of approximate coarse solves. In this paper, robustness of GenEO methods is analyzed with respect to approximate coarse solves. Interestingly, the GenEO-2 method has to be modified in order to be able to prove its robustness in this context.