Coarse spaces using extended generalized eigenproblems for heterogeneous Helmholtz problems

TL;DR

The work extends an abstract coarse-space framework based on extended generalized eigenproblems to heterogeneous Helmholtz problems, with analysis conducted at the continuous level. It introduces a two-level ORAS scheme where a coarse space is built from local eigenproblems and merged via a partition of unity, achieving stability when $\rho = \sigma \sqrt{k_{0} k_{1} \tau} < 1$. A key result is that local eigenvalues converge to zero, enabling effective coarse corrections that yield convergence improvements. Numerical experiments on the Marmousi model illustrate the expected trade-off: larger coarse spaces reduce iterations but increase setup time, guiding practical coarse-space design and filtering strategies.

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.

Figures (2)

• Figure 1: Velocity profile (top) and real part of the solution $u$ for $\omega=60\pi$ (bottom).
• Figure 2: GMRES iteration count (left) and total run time (right) with respect to $\omega$.

Theorems & Definitions (4)

• proposition 1
• proof
• theorem 1
• proof