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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

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
Full Text: DOI EuDML
[1] R. S. ANDERSSEN, J. R. CLEARY, Asymptotic Structure in Torsional Free Oscillations of Earth I. Geophys. J. R. Astr. Soc., 39, 1974, 241-268. Zbl0365.73095 · Zbl 0365.73095
[2] I. BABUSKA, J. E. OSBORN, Numerical Treatment of Eigenvalue Problems for Differential Equations with Discontinuons Coefficients. Math. Comp., 32, 1978, 991-1023. Zbl0418.65053 MR501962 · Zbl 0418.65053 · doi:10.2307/2006330
[3] [3] I BABUSKA, J. E. OSBORN, Analysis of Finite Element Methods for Second Order Boundary Value Problems using Mesh Dependent Norms. Numer. Math., 34, 1980, 41-62. Zbl0404.65055 MR560793 · Zbl 0404.65055 · doi:10.1007/BF01463997 · eudml:132658
[4] I. BABUSKA, J. E. OSBORN, Generalized Finite Element Methods : Their Performance and Their Relation to Mixed Methods. SIAM J. Numer. Anal., 20, 1983, 510-536. Zbl0528.65046 MR701094 · Zbl 0528.65046 · doi:10.1137/0720034
[5] [5] U. BANERJEE, Lower Norm Error Estimates for Approximate Solutions of Differential Equations with Non-Smooth Coefficients. Numer. Math, 51, 1987,303-321. Zbl0613.65087 MR895089 · Zbl 0613.65087 · doi:10.1007/BF01400117 · eudml:133197
[6] U. BANERJEE, Approximation of Eigenvalues of Differential Equations with Rough Coefficients. Ph. D. thesis, 1985, Univ. of Md., College Park, MD 20742.
[7] J. H. BRAMBLE, J. E. OSBORN, Rate of Convergence Estimate for Non-Selfadjoint Eigenvalue Approximations. Math. Comp., 27, 1973, 523-549. Zbl0305.65064 MR366029 · Zbl 0305.65064 · doi:10.2307/2005658
[8] F. CHATELIN, Spectral Approximation of Linear Operators, Academia Press, 1983. Zbl0517.65036 MR716134 · Zbl 0517.65036
[9] [9] R. S. FALK, J. E. OSBORN, Error Estimates for Mixed Methods, R.A.I.R.O. Numer. Anal. 14, 1980, 249-277. Zbl0467.65062 MR592753 · Zbl 0467.65062 · eudml:193361
[10] S. K. GARG, V. SVALBONAS and G. A. GURTMAN, Analysis of structural Composite Materials, Marcel Dekker, NY, 1973.
[11] E. R. LAPWOOD, The Effect of Discontinuities in Density and Rigidity on Torsional Eigenfrequencies. Geophys. J. R. Astr. Soc., 1975, 40, 453-464. Zbl0297.73064 · Zbl 0297.73064
[12] S. NEMAT-NASSER, General Variational Methods for Elastic Waves in Composities. J. Elasticity, 2, 1972, 73-90.
[13] S. NEMAT-NASSER, General Variational Principles in Nonlinear and Linear Elasticity with Applications. Mechanics Today, 1, 1974, 214-261. Zbl0305.73007 · Zbl 0305.73007
[14] S. NEMAT-NASSER, F. FU, Harmonic Waves in Layered Composites ; Bounds on Eigenfrequencies, J. Appl. Mech., 41, 1974, 288-290. Zbl0296.73025 · Zbl 0296.73025 · doi:10.1115/1.3423245
[15] J. R. OSBORN, Spectral Approximation of Compact Operators. Math. Comp., 29, 1975, 712-725. Zbl0315.35068 MR383117 · Zbl 0315.35068 · doi:10.2307/2005282
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