GenEO coarse spaces for heterogeneous indefinite elliptic problems

TL;DR

This paper compares spectral coarse spaces of GenEO type for heterogeneous indefinite elliptic problems, showing that one formulation offers more robustness in highly indefinite cases, supported by numerical experiments and ongoing theoretical development.

Contribution

It introduces and compares two formulations of spectral coarse spaces for indefinite elliptic problems, highlighting the superior robustness of one formulation in challenging scenarios.

Findings

One coarse space formulation is more robust for highly indefinite problems.

Numerical experiments confirm similar performance for mildly indefinite cases.

Theoretical analysis for one formulation is underway.

Abstract

Motivated by recent work on coarse spaces for Helmholtz problems, we provide in this paper a comparative study on the use of spectral coarse spaces of GenEO type for heterogeneous indefinite elliptic problems within an additive overlapping Schwarz method. In particular, we focus here on two different but related formulations of local generalised eigenvalue problems and compare their performance numerically. Even though their behaviour seems to be very similar for several well-known heterogeneous test cases that are mildly indefinite, only one of the coarse spaces has so far been analysed theoretically, while the other one leads to a significantly more robust domain decomposition method when the indefiniteness is increased. We present a summary of upcoming results developing such a theory and describe how the numerical experiments illustrate it.