Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

TL;DR

This paper extends the GenEO domain decomposition method to indefinite and non-self-adjoint convection-diffusion-reaction problems, providing theoretical iteration bounds and demonstrating practical efficiency for complex PDEs.

Contribution

It develops a new spectral coarse space for non-self-adjoint problems and proves GMRES iteration bounds that are robust against coefficient variations.

Findings

GMRES iteration counts are independent of diffusion coefficient heterogeneity.

The method remains effective despite increased non-self-adjointness and indefiniteness.

Practical tests show mild deterioration in convergence with problem complexity.

Abstract

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers which are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection-diffusion-reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalised eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts which are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the non-self-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus we obtain an iterative solver which is efficient in parallel and very effective for a wide range of convection--diffusion--reaction problems.