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.

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 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通过对连续亥姆霍兹问题应用紧凑扰动参数来研究，因此“足够丰富”的离散化导致连续问题的“足够小”扰动，并且通过 Fredholm 的替代方案继承了适定性. 然而，定性概念“足够丰富”涉及未知常数，并且仅具有渐近性质。我们的论文专注于一种完全离散的方法，通过模仿直接在离散级别上证明连续问题适定性的工具。这样，一个可计算的标准被推导出来，它证明了离散适定性，而不依赖于渐近扰动论证。通过使用这种新颖的方法，我们获得了 (a) 用于亥姆霍兹问题的 $hp$-FEM 的新存在性和唯一性结果，(b) 使得离散化变得不稳定的网格示例（Galerkin 矩阵是奇异的）和 (c)简单的检查算法 MOTZ 'marching-of-the-zeros'，它以后验的方式保证给定的网格被证明可以进行适定的亥姆霍兹离散化。