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

Abstract

Abstract. The Dirichlet eigenvalue or "drum" problem in a domain Ω ⊂ 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 √ 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( √ 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.

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

“…The main goal of this paper is to introduce a novel high-order boundary integral equation method for the numerical solution of (1) in the presence of a finite collection of punctures (1c). High-order methods for computing eigenvalues of the Laplacian and Helmholtz equations in two and three dimensions have been developed with domain decomposition methods [9,17,18], radial basis functions [43], boundary integral equations [6,45,20], the method of particular solution [7,25,33], the Dirichlet to Neumann map [8], and chebfun [19]. The method of fundamental solutions has also been used to compute eigenvalues of the biharmonic equation [40,3].…”

“…In Section 3.2, we describe an inexact Newton method to find the strength of the defects {α j } M j=1 and the eigenvalues λ. The decomposition (7) of the solution as the sum of a singular and regular part allows for the local behavior (4) to be precisely enforced while the regular part u R satisfies the homogeneous fourth-order PDE…”

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

“…To distinguish eigenvalues of multiplicity two as above, one could consider both Dirichlet and Neumann boundary conditions separately on thex-axis. For regions with smooth boundary, fundamental solutions have even proven to yield more accurate eigenvalues [39,40], however placement of the charge points is challenging for an a priori unknown and evolving domain.…”

“…For a given domain D 2 D, we use the method of particular solutions (MPS) to compute the D-L eigenvalues, K n (D) and eigenfunctions [36][37][38][39]. We choose two sets of particular solutions is an approximate D-L eigenvalue [38].…”

Estimates on Neumann eigenfunctions at the boundary, and the “method of particular solutions” for computing them

“…Indeed, this estimate implies that when |λ i − λ j | ≤ 1, then the inner product ψ i , ψ j is usually small compared with λ 2 . See Barnett [2].…”