Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant

TL;DR

This paper introduces a robust, efficient boundary integral method for solving the drum problem, achieving high accuracy and convergence using Nystrom approximation and Fredholm determinant techniques.

Contribution

It presents a novel approach combining Boyd's root-finding method with a combined field representation, enabling accurate eigenvalue computation for complex domains.

Findings

Achieves 13-digit accuracy in eigenvalue computations.

Demonstrates exponential convergence for analytic boundary domains.

Improves efficiency over traditional iterative methods.

Abstract

The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition-has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce two ideas to remedy this: 1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant. We show that this is many times faster than the usual iterative minimization of a singular value. 2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non-simply-connected domains. We implement the new method in two dimensions using spectrally accurate Nystrom product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including ones with strong exterior resonances.