Eigenvalue approximation by the finite element method: the method of Lagrange multipliers

Kolata, William G.

doi:10.1090/S0025-5718-1979-0514810-0

Eigenvalue approximation by the finite element method: the method of Lagrange multipliers
HTML articles powered by AMS MathViewer

by William G. Kolata PDF

Math. Comp. 33 (1979), 63-76 Request permission

Abstract:

The purpose of this paper is to investigate the application of the finite element method of Lagrange multipliers to the problem of approximating the eigenvalues of a selfadjoint elliptic operator satisfying Dirichlet boundary conditions. Although the Lagrange multiplier method is not a Rayleigh-Ritz-Galerkin approximation scheme, it is shown that at least asymptotically the Lagrange multiplier method has some of the properties of such a scheme. In particular, the approximate eigenvalues are greater than or equal to the exact eigenvalues and can be computed from a nonnegative definite matrix problem. It is also shown that the known estimates for the eigenvalue error are optimal.

References

Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179–192. MR 359352, DOI 10.1007/BF01436561

The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations

J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525–549. MR 366029, DOI 10.1090/S0025-5718-1973-0366029-9
James H. Bramble and Alfred H. Schatz, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math. 23 (1970), 653–675. MR 267788, DOI 10.1002/cpa.3160230408
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
George J. Fix, Eigenvalue approximation by the finite element method, Advances in Math. 10 (1973), 300–316. MR 341900, DOI 10.1016/0001-8708(73)90113-8
George M. Fix, Hybrid finite element methods, SIAM Rev. 18 (1976), no. 3, 460–484. MR 416066, DOI 10.1137/1018077
William G. Kolata, Approximation in variationally posed eigenvalue problems, Numer. Math. 29 (1977/78), no. 2, 159–171. MR 482047, DOI 10.1007/BF01390335
Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI 10.1007/BF02995904
John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI 10.1090/S0025-5718-1975-0383117-3