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External finite-element approximations of eigenfunctions in the case of multiple eigenvalues. (English) Zbl 0811.65090
The paper deals with the finite element analysis of second-order elliptic eigenvalue problems when the domains \(\Omega_ h\) are not contained in the original domain \(\Omega\subset \mathbb{R}^ 2\). The main aim of the paper is to study the convergence of approximate eigenfunctionals in the case of multiple exact eigenvalues, extending an approach formerly established by the authors in the case of simple eigenvalues. It differs from the approach of I. Babuška and J. E. Osborn [Math. Comput. 52, No. 186, 275-297 (1989; Zbl 0675.65108)] in that no use is made of operator theory.

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
Full Text: DOI
[1] Babuška, I.; Osborn, J.E., Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues, SIAM J. numer. anal., 24, 1249-1276, (1987) · Zbl 0701.65042
[2] Babuška, I.; Osborn, J.E., Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. comp., 52, 275-297, (1989) · Zbl 0675.65108
[3] Babuška, I.; Osborn, J.E., Eigenvalue problems, (), 641-787 · Zbl 0875.65087
[4] Banerjee, U.; Osborn, J.E., Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. math., 56, 735-762, (1990) · Zbl 0693.65071
[5] Ciarlet, P.G., The finite element method for elliptic problem, () · Zbl 0285.65072
[6] Dautray, R.; Lions, J.-L., Analyse mathématique et calcul numérique pour LES sciences et LES techniques, tome 2, (1985), Masson Paris
[7] Glowinski, R.; Lions, J.-L.; Trémolieres, R., Analyse numérique des inéquations variationelles, (1976), Dunod Paris · Zbl 0358.65091
[8] Kufner, A.; John, O.; Fučik, S., Function spaces, (1977), Noordhoff Leiden
[9] Mikhlin, S.G., Partielle differentialgleichungen in der mathematischen physik, (1978), Akademie Verlag Berlin · Zbl 0447.41006
[10] Nečas, J., LES Méthodes directes en théorie des équations elliptiques, (1967), Masson Paris · Zbl 1225.35003
[11] Oganesian, L.A.; Rukhovec, L.A., LES Méthodes directes en théorie des équations elliptiques, (1979), Akad. Nauk Armyan, SSR Erevan, (in Russian)
[12] Raviart, P.A.; Thomas, J.M., LES Méthodes directes en théorie des équations elliptiques, (1983), Masson Paris
[13] Vanmaele, M., Contribution to the theory of finite element methods for second order eigenvalue problems, () · Zbl 0802.65106
[14] Vanmaele, M.; Van Keer, R., Error estimates for a finite element method with numerical quadrature for a class of elliptic eigenvalue problems, (), 267-282 · Zbl 0760.65096
[15] Vanmaele, M.; Ženíšek, A., External finite element approximations of eigenvalue problems, RAIRO modél. math. anal. numér., 27, 565-589, (1993) · Zbl 0792.65086
[16] Ženíšek, A., Nonlinear elliptic and evolution problems and their finite element approximations, (1990), Academic Press London · Zbl 0731.65090
[17] Zlámal, M., Curved elements in the finite element method I, SIAM J. numer. anal., 10, 229-240, (1973) · Zbl 0285.65067
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