Unique continuation principles for finite-element discretizations of the Laplacian
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| Titel: | Unique continuation principles for finite-element discretizations of the Laplacian |
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| Autoren: | Graham Cox, Scott MacLachlan, Luke Steeves |
| Quelle: | Linear Algebra and its Applications. 727:84-111 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Elsevier BV, 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Spectral Theory (math.SP), Analysis of PDEs (math.AP) |
| Beschreibung: | Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations are a natural tool to help gain understanding into elliptic equations, it is natural to ask if such principles also hold at the discrete level. In this work, we prove a version of the unique continuation principle for piecewise-linear and -bilinear finite-element discretizations of the Laplacian eigenvalue problem on polygonal domains in $\mathbb{R}^2$. Namely, we show that any solution to the discretized equation $-Δu = λu$ with vanishing Dirichlet and Neumann traces must be identically zero under certain geometric and topological assumptions on the resulting triangulation. We also provide a counterexample, showing that a nonzero \emph{inner solution} exists when the topological assumptions are not satisfied. Finally, we give an application to an eigenvalue interlacing problem, where the space of inner solutions makes an explicit appearance. |
| Publikationsart: | Article |
| Sprache: | English |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2025.07.029 |
| DOI: | 10.48550/arxiv.2410.08963 |
| Zugangs-URL: | http://arxiv.org/abs/2410.08963 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Dokumentencode: | edsair.doi.dedup.....5dfa180cbe3d0149e34c9ed6d7079199 |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations are a natural tool to help gain understanding into elliptic equations, it is natural to ask if such principles also hold at the discrete level. In this work, we prove a version of the unique continuation principle for piecewise-linear and -bilinear finite-element discretizations of the Laplacian eigenvalue problem on polygonal domains in $\mathbb{R}^2$. Namely, we show that any solution to the discretized equation $-Δu = λu$ with vanishing Dirichlet and Neumann traces must be identically zero under certain geometric and topological assumptions on the resulting triangulation. We also provide a counterexample, showing that a nonzero \emph{inner solution} exists when the topological assumptions are not satisfied. Finally, we give an application to an eigenvalue interlacing problem, where the space of inner solutions makes an explicit appearance. |
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| ISSN: | 00243795 |
| DOI: | 10.1016/j.laa.2025.07.029 |