Solvability of discrete Helmholtz equations

Abstract

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 argument 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 existence and uniqueness results for the $hp$-FEM for the Helmholtz problem, (b) examples for meshes such that the discretization becomes unstable (Galerkin 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 well-posed Helmholtz discretization.