Solving the unique continuation problem for Schrödinger equations with low regularity solutions using a stabilized finite element method
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908337971396608 |
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| author | Burman, Erik Lu, Mingfei Oksanen, Lauri |
| author_facet | Burman, Erik Lu, Mingfei Oksanen, Lauri |
| contents | In this paper, we consider the unique continuation problem for the Schrödinger equations. We prove a Hölder type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving the unique continuation problem for Schrödinger equations with low regularity solutions using a stabilized finite element method Burman, Erik Lu, Mingfei Oksanen, Lauri Numerical Analysis In this paper, we consider the unique continuation problem for the Schrödinger equations. We prove a Hölder type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory. |
| title | Solving the unique continuation problem for Schrödinger equations with low regularity solutions using a stabilized finite element method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.16914 |