{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/solvability-of-discrete-helmholtz-equations","title":"Solvability of Discrete Helmholtz Equations","arxiv_id":"2105.02273","date":"2021-05-05","proceeding":null,"authors":["Maximilian Bernkopf","Stefan Sauter","Céline Torres","Alexander Veit"],"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.","url_abs":"https://arxiv.org/abs/2105.02273v2","url_pdf":"https://arxiv.org/pdf/2105.02273v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"solvability-of-discrete-helmholtz-equations","repo_url":"https://github.com/alexander-veit/motz","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}