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Computing lower eigenvalues on rough domains

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If you have a question about this talk, please contact Nicolas Boulle.

In this talk I will describe a strategy for finding sharp upper and lower numerical bounds of the Poincare constant on a class of planar domains with piecewise self-similar boundary. The approach is developed in [A] and it consists of four main blocks: 1) tight inner-outer shape interpolation, 2) conformal mapping of the approximate polygonal regions, 3) grad-div system formulation of the spectral problem and 4) computation of the eigenvalue bounds. After describing the method, justifying its validity and reporting on general convergence estimates, I will show concrete evidence of its effectiveness on the Koch snowflake. I will conclude the talk by discussing potential applications to other linear operators on rough regions. This research has been conducted jointly with Lehel Banjai (Heriot-Watt University).

[A] J. Fractal Geometry 8 (2021) No. 2, pp. 153-188

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

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