Solvability of Discrete Helmholtz Equations
Abstract
We study the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin -finite element method. Well-posedness of the discrete equations is typically investigated by applying a compact perturbation to the continuous Helmholtz problem so that a "sufficiently rich" discretization results in a "sufficiently small" perturbation of the continuous problem and well-posedness is inherited via Fredholm's alternative. The qualitative notion "sufficiently rich", however, involves unknown constants and is only of asymptotic nature. Our paper is focussed on a fully discrete approach by mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level. In this way, a computable criterion is derived which certifies discrete well-posedness without relying on an asymptotic perturbation argument. By using this novel approach we obtain a) new stability results for the -FEM for the Helmholtz problem b) examples for meshes such that the discretization becomes unstable (stiffness matrix is singular), and c) a simple checking Algorithm MOTZ "marching-of-the-zeros" which guarantees in an a posteriori way that a given mesh is certified for a stable Helmholtz discretization.
Cite
@article{arxiv.2105.02273,
title = {Solvability of Discrete Helmholtz Equations},
author = {Maximilian Bernkopf and Stefan Sauter and Céline Torres and Alexander Veit},
journal= {arXiv preprint arXiv:2105.02273},
year = {2022}
}