Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

Details Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

2012-03-24

arxiv.org/abs/1203.5444v1

Mathematics

Differential Geometry

7 Pages, No Figures

Scientific paper

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.

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