  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.
