Computing eigenvalues of the Laplacian on rough domains
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- by Frank Rösler and Alexei Stepanenko
- Math. Comp. 93 (2024), 111-161
- DOI: https://doi.org/10.1090/mcom/3827
- Published electronically: May 10, 2023
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Abstract:
We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.References
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Bibliographic Information
- Frank Rösler
- Affiliation: Department of Mathematics, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland
- ORCID: 0000-0002-7431-3961
- Email: frank.roesler@unibe.ch
- Alexei Stepanenko
- Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, Wales, United Kingdom
- MR Author ID: 1416209
- Email: StepanenkoA@cardiff.ac.uk
- Received by editor(s): April 22, 2021
- Received by editor(s) in revised form: July 1, 2022, January 10, 2023, and April 5, 2023
- Published electronically: May 10, 2023
- Additional Notes: The first author was supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No 885904. The research of the second author was supported by the United Kingdom Engineering and Physical Sciences Research Council, through its Doctoral Training Partnership with Cardiff University.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 111-161
- MSC (2020): Primary 47F10, 35P15, 65N25
- DOI: https://doi.org/10.1090/mcom/3827