On a variational approximation method for a class of elliptic eigenvalue problems in composite structures

Vanmaele, M.; Keer, R.

doi:10.1090/S0025-5718-96-00741-7

On a variational approximation method for a class of elliptic eigenvalue problems in composite structures
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by M. Vanmaele and R. Van Keer PDF

Math. Comp. 65 (1996), 999-1017 Request permission

Abstract:

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in $M$ nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalent variational formulation in an abstract setting in product Hilbert spaces. We use standard Lagrange finite element spaces on the subdomains. Moreover, the bilinear forms are approximated by suitable numerical quadrature formulas. We obtain error estimates for both the eigenfunctions and the eigenvalues, allowing for the case of multiple exact eigenvalues, by a pure variational method.

References

A. B. Andreev, V. A. Kascieva, and M. Vanmaele, Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas, J. Comput. Appl. Math. 43 (1992), no. 3, 291–311. MR 1193808, DOI 10.1016/0377-0427(92)90016-Q
Uday Banerjee and John E. Osborn, Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. Math. 56 (1990), no. 8, 735–762. MR 1035176, DOI 10.1007/BF01405286
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1, Collection du Commissariat à l’Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1984 (French). With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean-Michel Combes, André Gervat, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily. MR 792484
Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 2, Collection du Commissariat à l’Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1985 (French). With the collaboration of Michel Artola, Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard and Bruno Scheurer. MR 902801
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448814
J. Kačur & R. Van Keer , On the Numerical Solution of some Heat Transfer Problems in Multi-component Structures with nonperfect Thermal Contacts. In: R.W. Lewis, (editor), Numerical Methods for Thermal Problems VII, Pineridge Press, Swansea (1991) 1378–1388.
H. Kardestuncer and D. H. Norrie (eds.), Finite element handbook, McGraw-Hill Book Co., New York, 1987. MR 900813
Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. MR 0482102
M.D. Mikhailov & M.N. Özişik , Unified Analysis and Solutions of Heat and Mass Diffusion, John Wiley & Sons, N.Y. (1984).
M.N. Özişik , Heat Conduction, (2nd edition), John Wiley & Sons, N.Y. (1993).
A.W. Pratt , Heat Transmission in Buildings, J. Wiley, Chichester (1981).
P.-A. Raviart and J.-M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). MR 773854
M. Vanmaele , A numerical quadrature finite element method for 2nd-order eigenvalue problems with Dirichlet-Robin boundary conditions, Proceedings ISNA’92, Prague (1994), 269–292.
M. Vanmaele & R. Van Keer , Convergence and error estimates for a finite element method with numerical quadrature for a second-order elliptic eigenvalue problem, Int. Series of Num. Math., 96 (1991) 225–236.
M. Vanmaele and A. Ženíšek, External finite-element approximations of eigenfunctions in the case of multiple eigenvalues, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 51–66. MR 1284251, DOI 10.1016/0377-0427(94)90289-5