Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

The Dirichlet eigenvalue or “drum” problem in a domain $\Omega\subset\mathbb{R}^2$ becomes numerically challenging at high eigenvalue (frequency) $E$. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as $\sqrt{E}$, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is $E$. Our main result is an inclusion bound on eigenvalues that is a factor $O(\sqrt{E})$ tighter than the classical bound of Moler-Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort.