Computing ultra-precise eigenvalues of the Laplacian within polygons

TL;DR

This paper extends classical methods to compute eigenvalues of the Laplacian in polygons with unprecedented precision, often exceeding a thousand digits, by exploiting symmetry and eigenfunction expansion properties.

Contribution

It demonstrates a high-precision eigenvalue computation technique for polygons with non-analytic vertices, including methods for bounding eigenvalues through eigenfunction expansion alternation.

Findings

Eigenvalues computed to over a thousand digits.

Alternation in eigenvalue approximations provides bounds.

Method applies to polygons with re-entrant angles and various boundary conditions.

Abstract

The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, very often to well over a hundred digits, and sometimes to over a thousand digits. A curious observation is that as one increases the number of terms in the eigenfunction expansion, the approximate eigenvalue may be made to alternate above and below the exact eigenvalue. This alternation provides a new method to bound eigenvalues, by inspection. Symmetry must be exploited to simplify the geometry, reduce the number of non-analytic vertices and disentangle degeneracies. The symmetry-reduced polygons considered here have at most one non-analytic vertex from which all edges can be seen. Dirichlet, Neumann, and periodic-type edge conditions, are independently imposed on each polygon edge. The full shapes include the regular polygons and some with re-entrant angles (cut-square, L-shape, 5-point star). Thousand-digit results are obtained for the lowest Dirichlet eigenvalue of the L-shape, and regular pentagon and hexagon.