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Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem. (English) Zbl 1474.65253

Summary: 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.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

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