Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem

An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal. Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods. The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems. The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one. As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.