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Approximation of eigenvalues of differential equations with non-smooth coefficients. (English) Zbl 0639.65051
This paper deals with the approximation of eigenvalues and eigenfunctions of the problem \(-(au')'+cu=\lambda bu\), \(u(0)=u(1)=0\) where the coefficients a, b, c are non-smooth functions of bounded variations via the \(L_ 2\) finite element method (FEM) introduced by I. Babuska and J. E. Osborn [SIAM J. Numer. Anal. 20, 510-536 (1983; Zbl 0528.65046)]. In Sections 2 and 3 some notions and notations and known results of spectral approximation are given. The \(L_ 2\) FEM is introduced in Section 4 along with two finite dimensional subspaces. Then error estimates for the approximate eigenvectors and eigenvalues are derived. Numerical results are also given. The conclusions are: the approximate eigenvalues obtained from the \(L_ 2\) FEM are more accurate than those obtained from the standard FEM for these problems, the computational effort involved in the \(L_ 2\) FEM is the same as the standard FEM and thus should be prefered for problems with non-smooth coefficients.
Reviewer: Z.Schneider

MSC:
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34L99 Ordinary differential operators
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