Computing ultra-precise eigenvalues of the Laplacian within polygons
This work provides ultra-precise eigenvalue benchmarks for polygonal domains, which are useful for validating numerical methods in spectral geometry and computational physics.
The authors extend the Fox-Henrici-Moler method to compute eigenvalues of the Laplacian within polygons to unprecedented precision, often over 100 digits and sometimes over 1000 digits. They achieve thousand-digit results for the lowest Dirichlet eigenvalue of the L-shape, regular pentagon, and regular hexagon.
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.