Physical Sciences and Mathematics Commons^™

Perturbative Analysis Of The Method Of Particular Solutions For Improved Inclusion Of High-Lying Dirichlet Eigenvalues, A. H. Barnett

Dartmouth Scholarship

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 …