High-Accuracy Calculation of Eigenvalues of the Laplacian in an Ellipse (with Neumann Boundary Condition)

Abstract

A numerical technique for solving the eigenvalue problem for the Laplacian in an ellipse is described. The results are based on K.I. Babenko’s ideas. In elliptic coordinates, the variables in the Laplace equation for an ellipse are separated and the problem of calculating the eigenvalues is reduced to the study of Mathieu functions. The integral in the variational principle is computed using a global quadrature rule. The minimization of a quadratic functional is reduced to the minimization of a quadratic form, which leads to an algebraic eigenvalue problem.