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Resonant modes of an elliptic membrane are computed for a wide range of frequencies using a Galerkin formulation. Results are confirmed using Mathieu functions and finite-element methods. Algorithms and their implementations are described to handle Dirichlet or Neumann boundary conditions and draw animations or contour plots of the modal surfaces. The methods agree to four or more digit accuracy for the first one hundred modes. The effects of high function order and high frequency parameter upon the convergence of the modified Mathieu function series are discussed and quantified. The Galerkin method is conceptually simple and requires only an eigenvalue solver without the need of special functions.

Laplace's equation is solved via separation of variables in toroidal coordinates

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