On the Convergence of the Fourier Approximation for Eigenvalues and Eigenfunctions of Discontinuous Problems

In this paper, we consider a model eigenvalue problem with discontinuous coefficients in order to study the convergence of the Fourier methods applied to this problem. We prove that the rate of convergence of the Fourier-Galerkin method is third order for the eigenvalues and order 2.5 for the eigenfunctions. For the Fourier collocation method we obtained only second order accuracy. We also show that the Fourier collocation method can be improved by a preprocessing of the coefficients. The theory is confirmed by numerical results.