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Eigenvalue approximation by the finite element method: The method of Lagrange multipliers. (English) Zbl 0448.65067


MSC:

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
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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[1] Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179 – 192. · Zbl 0258.65108
[2] I. BABUŠKA & A. K. AZIZ, ”Survey lectures on the mathematical foundations of the finite element method,” The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1973, pp. 5-359.
[3] J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525 – 549. · Zbl 0305.65064
[4] 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. · Zbl 0204.11102
[5] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. · Zbl 0128.34803
[6] George J. Fix, Eigenvalue approximation by the finite element method, Advances in Math. 10 (1973), 300 – 316. · Zbl 0257.65086
[7] George M. Fix, Hybrid finite element methods, SIAM Rev. 18 (1976), no. 3, 460 – 484. · Zbl 0332.35005
[8] William G. Kolata, Approximation in variationally posed eigenvalue problems, Numer. Math. 29 (1977/78), no. 2, 159 – 171. · Zbl 0352.65059
[9] W. G. KOLATA, ”On solving a matrix problem arising in a hybrid finite element method.” (Preprint.) · Zbl 1187.91170
[10] Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). · Zbl 1225.35003
[11] J. NITSCHE, ”A projection method for Dirichlet problems using subspaces with nearly zero boundary conditions.” (Preprint.) · Zbl 0271.65059
[12] 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). Collection of articles dedicated to Lothar Collatz on his sixtieth birthday. · Zbl 0229.65079
[13] John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712 – 725. · Zbl 0315.35068
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