Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu’s system, Lame’s system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems. Keywords: Helmholtz equation, Schrödinger equation, separation of variables, Mathieu’s system, Lame’s system, spectral methods, Chebyshev collocation, Laguerre collocation, multiparameter eigenvalue problem, two-parameter eigenvalue problem, three-parameter eigenvalue problem, Sylvester equation, Bartels-Stewart method, subspace methods