Solvability of Discrete Helmholtz Equations

TL;DR

This paper develops a new, computable criterion for certifying the well-posedness of discretized Helmholtz equations using $hp$-finite element methods, enabling stability verification and identifying unstable meshes.

Contribution

It introduces a fully discrete approach that directly assesses stability without asymptotic perturbation, providing new stability results and an algorithm for mesh certification.

Findings

Derived a computable stability criterion for discrete Helmholtz problems.

Identified mesh configurations leading to discretization instability.

Developed the MOTZ algorithm for a posteriori mesh certification.

Abstract

We study the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin $hp$-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 $hp$-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.