Computing eigenvalues of the Laplacian on rough domains

Frank Rösler; Alexei Stepanenko

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arXiv · 2104.09444

Computing eigenvalues of the Laplacian on rough domains

Frank Rösler·Alexei Stepanenko

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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 Poincar\'e-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 counter example 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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BibTeX RIS

Frank Rösler, Alexei Stepanenko. 2021-04-19. Computing eigenvalues of the Laplacian on rough domains. https://arxiv.org/abs/2104.09444

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