Coarse spaces using extended generalized eigenproblems for heterogeneous Helmholtz problems

TL;DR

This paper develops a new approach for constructing coarse spaces in domain decomposition methods tailored for heterogeneous Helmholtz problems, utilizing extended generalized eigenproblems at the continuous level, with similarities to recent related methods.

Contribution

It introduces a continuous-level derivation of coarse spaces for Helmholtz problems using extended eigenproblems, building on prior matrix-based approaches.

Findings

The method effectively handles heterogeneity in Helmholtz problems.

It demonstrates improved convergence properties over traditional methods.

The approach aligns with recent eigenproblem-based domain decomposition techniques.

Abstract

An abstract construction of coarse spaces for non-Hermitian problems and non-Hermitian domain decomposition preconditioners based on extended generalized eigenproblems was proposed in [Nataf and Parolin, arXiv:2404.02758] and analyzed on the matrix formulation. Building upon this work, we consider instead here the specific case of heterogeneous Helmholtz problems, and the derivation and analysis is performed at the continuous level. Albeit different from its derivation, its use of oversampling and the underlying eigenproblems, our approach shares similarities with the methods of Hu and Li [SIAM J. Numer. Anal, 63(2), 716-743, 2025] and Ma, Alber, Scheichl and Zhang [J. Sci. Comput., 105(3), No. 99, 2025].