Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

TL;DR

This paper introduces an analytical level set method for highly accurate computation of Laplace eigenvalues on smooth domains by representing eigenfunctions as linear combinations of Bessel functions and adjusting parameters to fit the domain boundary.

Contribution

The paper presents a novel analytical level set approach that achieves high-precision Laplace eigenvalue computations for smooth domains, especially effective for domains like ellipses.

Findings

Coefficients decay exponentially for certain domains

Method achieves arbitrarily high accuracy for some shapes

Effective for smooth domains with modest eccentricity

Abstract

This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction $u$ as a linear combination of eigenfunctions corresponding to the common eigenvalue $\rho ^{2}$:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,\theta) =\sum_{n=0}^{N}P_{n}J_{n}(\rho) \cos n\theta,}}{1}{}{}{}}We adjust the coefficients $P_{n}$ and the parameter $\rho $ so that the zero level set of $u$ approximates the domain of interest. For some domains, such as ellipses of modest eccentricity, the coefficients $P_{n}$ decay exponentially and the proposed method can be used to compute eigenvalues with arbitrarily high accuracy.