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On the convergence of eigenvalues for mixed formulations. (English) Zbl 1003.65052
Summary: Eigenvalue problems for mixed formulation show peculiar features that make them substantially different from the corresponding mixed direct problems. In this paper we analyze, in an abstract framework, necessary and sufficient conditions for their convergence.

MSC:
65J10 Numerical solutions to equations with linear operators
47A75 Eigenvalue problems for linear operators
35P15 Estimates of eigenvalues in context of PDEs
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
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References:
[1] I BabuŠka , Error-Bounds for Finite Element Method , Numer. Math. 16 ( 1971 ), 322 - 333 . Article | MR 288971 | Zbl 0214.42001 · Zbl 0214.42001
[2] I Babuška , On the finite element method with Lagrangian multipliers , Numer. Math. 20 ( 1973 ), 179 - 192 . Article | MR 359352 | Zbl 0258.65108 · Zbl 0258.65108
[3] I Babuška - J.E. Osborn , ” Handbook of Numerical Analysis ”, vol. II , ch. Eigenvalue Problems, North-Holland , 1991 , pp. 641 - 788 . MR 1115240 | Zbl 0875.65087 · Zbl 0875.65087
[4] K.J. Bathe - C. Nitikitpaiboon - X. Wang , A mixed displacement-based finite element formulation for acoustic fluid-structure interaction, Computers & Structures 56 ( 1995 ), 225 - 237 . MR 1336298 | Zbl 1002.76536 · Zbl 1002.76536
[5] D. Boffi - F. Brezzi - L. Gastaldi , On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form , submitted to Math. Comp. , 1997 . MR 1642801 | Zbl 0938.65126 · Zbl 0938.65126
[6] J.M. Boland - R. Nicolaides , On the stability of bilinear-constant velocity-pressure finite elements , Numer. Math. 44 ( 1984 ), 219 - 222 . Article | MR 753954 | Zbl 0544.76030 · Zbl 0544.76030
[7] J.H. Bramble - J.E. Osborn , Rate of convergence for nonselfadjoint eigenvalue approximations , Math. Comp. 27 ( 1973 ), 525 - 549 . MR 366029 | Zbl 0305.65064 · Zbl 0305.65064
[8] F. Brezzi , On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers , R.A.I.R.O. Anal. Numer. 8 ( 1974 ), 129 - 151 . Numdam | MR 365287 | Zbl 0338.90047 · Zbl 0338.90047
[9] F. Brezzi - M. Fortin , ” Mixed and Hybrid Finite Element Methods ”, Springer-Verlag , New York , 1991 . MR 1115205 | Zbl 0788.73002 · Zbl 0788.73002
[10] F. Brezzi - J. Douglas Jr. - M. Fortin - L.D. Marini , Efficient rectangular mixed finite elements in two and three space variables , R.A.I.R.O. Model. Math. Anal. Numer. 21 ( 1987 ), 237 - 250 . Numdam | MR 921828 | Zbl 0689.65065 · Zbl 0689.65065
[11] F. Brezzi - J. Douglas Jr. - L.D. Marini , Two families of mixed finite elements for second order elliptic problems , Numer. Math. 47 ( 1985 ), 217 - 235 . Article | MR 799685 | Zbl 0599.65072 · Zbl 0599.65072
[12] P.G. Ciarlet , ” The Finite Element Method for Elliptic Problems ”, North-Holland , Amsterdam , 1978 . MR 520174 | Zbl 0383.65058 · Zbl 0383.65058
[13] P.G. Ciarlet - P.-A. Raviart , A mixed finite element methodfor the biharmonic equation , in ” Mathematical Aspects of Finite Element in Partial Differential Equations ”, C. de Boor (ed.), Academic Press , New York , 1974 , 125 - 143 . MR 657977 | Zbl 0337.65058 · Zbl 0337.65058
[14] J. Falk - J.E. Osborn , Error estimates for mixed methods , R.A.I.R.O. Anal. Numer. 4 ( 1980 ), 249 - 277 . Numdam | MR 592753 | Zbl 0467.65062 · Zbl 0467.65062
[15] M. Fortin , An analysis of the convergence of mixed finite element methods , R.A.I.R.O. Anal. Numer. 11 ( 1977 ), 341 - 354 . Numdam | MR 464543 | Zbl 0373.65055 · Zbl 0373.65055
[16] R. Glowinski , Approximations externes par éléments finis de Lagrange d’ ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthodes itératives de résolution des problèmes approchés, ”Topics in Numerical Analysis” , J. Miller (ed.), Academic Press , New York , 1973 , 123 - 171 . MR 351120 | Zbl 0277.35003 · Zbl 0277.35003
[17] P. Grisvard , ” Elliptic Problems in Non-Smooth Domains ”, Pitman , Marshfields, Mass ., 1985 . Zbl 0695.35060 · Zbl 0695.35060
[18] C. Johnson - J. Pitkäranta , Analysis of some mixed finite element methods related to reduced integration , Math. Comp. 38 ( 1982 ), 375 - 400 . MR 645657 | Zbl 0482.65058 · Zbl 0482.65058
[19] R.B. Kellogg - J.E. Osborn , A regularity result for the Stokes problem , J. Funct. Anal. 21 ( 1976 ), 397 - 431 . MR 404849 | Zbl 0317.35037 · Zbl 0317.35037
[20] B. Mercier , Numerical solution of the biharmonic problem by mixed finite elements of class C^\circ , Boll. U.M.I. 10 ( 1974 ), 133 - 149 . MR 378442 | Zbl 0332.65058 · Zbl 0332.65058
[21] B. Mercier - J.E. Osborn - J. Rappaz - P.-A. Raviart , Eigenvalue approximation by mixed and hybrid methods , Math. Comp. 36 ( 1981 ), 427 - 453 . MR 606505 | Zbl 0472.65080 · Zbl 0472.65080
[22] J.T. Oden - O. Jacquotte , Stability of some mixed finite element methods for Stokesian flows , Comp. Methods Appl. Mech. Eng. 43 ( 1984 ), 231 - 247 . MR 745509 | Zbl 0598.76033 · Zbl 0598.76033
[23] J.E. Osborn , Eigenvalue approximations by mixed methods , in ” Advances in Computer Methods for Partial Differential Equations III ”, R. Vichnevetsky and R. Stepleman (eds.), New Brunswick , 1979 , 158 - 161 . MR 603467
[24] P.-A. Raviart - J.M. Thomas , A mixed finite element method for second order elliptic problems , in ” Mathematical Aspects of the Finite Element Method ”, I. Galligani and E. Magenes (eds.), Lecture Notes in Math. , Springer-Verlag , New York , 1977 , 292 - 315 . MR 483555 | Zbl 0362.65089 · Zbl 0362.65089
[25] R. Scholz , A mixed method for fourth order problems using linear finite elements , R.A.I.R.O. Anal. Numer. 12 ( 1978 ), 85 - 90 . Numdam | MR 483557 | Zbl 0382.65059 · Zbl 0382.65059
[26] X. Wang - K.J. Bathe , On mixed elements for acoustic fluid-structure interaction , M3AS , 7 ( 1997 ), 329 - 344 . MR 1443789 | Zbl 0881.76053 · Zbl 0881.76053
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