Domain Decomposition Spectral Approximations for an Eigenvalue Problem with a Piecewise Constant Coefficient

Consider a model eigenvalue problem with a piecewise constant coefficient. We split the domain at the discontinuity of the coefficient function and define the multidomain variational formulation for the eigenproblem. The discrete multidomain variational formulations are defined for Legendre–Galerkin and Legendre-collocation methods. The spectral rate of convergence of the approximate eigensolutions is proven for the Legendre–Galerkin method. The minmax principle is used for the convergence analysis. The Legendre-collocation, Chebyshev-collocation, Legendre-collocation penalty, and Chebyshevcollocation penalty methods are also defined by using the multidomain approach, and their numerical results applied to the eigenproblem are demonstrated. The spectral convergence for the eigenvalues and eigenfunctions is confirmed for all the multidomain spectral techniques presented here.