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.

翻译：我们用符合的 Galerkin $hp$fite 元素法研究离散的离散性Helmholtz 问题与罗宾边界条件的独特溶解性。 离散性方程式的精密性通常通过对连续的Helmholtz 问题应用压缩扰动法来调查。 这样, “ 足够丰富” 的离散性导致连续问题“ 足够小” 的“ 足够小” 扰动性, 并通过 Fredholm 的替代法来继承。 但是, 质量概念“ 足够丰富 ”, 涉及未知的常数, 并且只是无药性。 我们的纸张侧重于完全离散的方法, 将工具模拟用于直接在离散水平上证明持续问题的正确性的工具。 这样, 一个可比较性标准可以证明离散性, 而不依赖一个无症状的扰动性扰动性 。 我们通过这种新方法获得一种新的稳定性结果, 用于给定的 Helmtz 问题中的 美元- FEM, 是一个稳定性磁性 的模型模型, 。