Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

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.