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Computing eigenvalues of the Laplacian on rough domains

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In this talk, I shall present a joint work with Frank Rösler in which we consider the computability of eigenvalues of the Dirichlet Laplacian on bounded domains with rough, possibly fractal, boundaries. We work within the framework of Solvability Complexity Indices, which allows us to formulate the problem rigorously. On one hand, we construct an algorithm that provably converges for any domain satisfying a collection of mild topological hypotheses, on the other, we prove that there does not exist an algorithm of the same type which converges for an arbitrary bounded domain. Along the way, we develop new spectral convergence results for the Dirichlet Laplacian on rough domains, as well as a novel Poincaré-type inequality.

This talk is part of the Applied and Computational Analysis series.

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