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.

中文翻译：

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离散亥姆霍兹方程的可解性

我们使用一致的Galerkin $ hp $-有限元方法研究了具有Robin边界条件的离散亥姆霍兹问题的独特可解性。离散方程的适定性通常是通过对连续Helmholtz问题应用紧致扰动来研究的，这样“足够丰富”的离散化会导致连续问题的“足够小”扰动，而适定性是通过Fredholm的替代方法继承的。但是，定性概念“足够丰富”涉及未知常数，并且仅具有渐近性质。我们的论文集中在完全离散的方法上，方法是模仿用于直接在离散级别上证明连续问题的适定性的工具。这样，得出一个可计算的标准，该标准可证明不依赖于渐近摄动论点的离散适定性。通过使用这种新颖的方法，我们获得了a）亥姆霍兹问题的$ hp $ -FEM的新稳定性结果b）网格示例，使得离散化变得不稳定（刚度矩阵是奇异的），并且c）简单检查了算法MOTZ“零进军”，以后验方式确保给定的网格已通过稳定的亥姆霍兹离散化认证。