Computing eigenvalues of the Laplacian on rough domains

Rösler, Frank; Stepanenko, Alexei

doi:10.1090/mcom/3827

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.

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