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Johnson, Claes; Pitkäranta, Juhani

Analysis of some mixed finite element methods related to reduced integration. (English) Zbl 0482.65058

Math. Comput. 38, 375-400 (1982).
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Cited in 4 Reviews
Cited in 50 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74K20 Plates
76D05 Navier-Stokes equations for incompressible viscous fluids
65N15 Error bounds for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics

Keywords:

mixed finite element methods; reduced integration; error estimates; Stokes’ problem; plate problem; Babuska-Brezzi type stability estimate
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References:

[1] Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322 – 333. · Zbl 0214.42001 · doi:10.1007/BF02165003
[2] I. Babuška, J. Osborn & J. Pitkäranta, Analysis of Mixed Methods Using Mesh Dependent Norms, Report #2003, Mathematics Research Center, University of Wisconsin, 1979. · Zbl 0472.65083
[3] M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér. 12 (1978), no. 3, 211 – 236, iii (English, with French summary). · Zbl 0428.65059
[4] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129 – 151 (English, with loose French summary). · Zbl 0338.90047
[5] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[6] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33 – 75. · Zbl 0302.65087
[7] V. Girault, A combined finite element and marker and cell method for solving Navier-Stokes equations, Numer. Math. 26 (1976), no. 1, 39 – 59. · Zbl 0313.65105 · doi:10.1007/BF01396565
[8] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. · Zbl 0413.65081
[9] R. Glowinski and O. Pironneau, On numerical methods for the Stokes problem, Energy methods in finite element analysis, Wiley, Chichester, 1979, pp. 243 – 264. · Zbl 0415.76024
[10] V. A. Kondrat\(^{\prime}\)ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209 – 292 (Russian).
[11] D. Malkus & T. Hughes, ”Mixed finite element methods–Reduced and selective integration techniques: A unification of concepts,” Comput. Methods Appl. Mech. Engrg., v. 15, 1978, pp. 63-81. · Zbl 0381.73075
[12] H. Melzer & R. Rannacher, Spannungskonzentrationen in Eckpunkten der vertikalen belasteten Kirchoffschen Platte, Universität Bonn, 1979. (Preprint.)
[13] R. L. Sani, P. M. Gresho & R. L. Lee, On the Spurious Pressures Generated by Certain GFEM Solutions of the Incompressible Navier-Stokes Equations, Technical report, Lawrence Livermore Laboratory, Oct. 1979. · Zbl 0446.76034
[14] Ranbir S. Sandhu and Kamar J. Singh, Reduced integration for improved accuracy of finite element approximations, Comput. Methods Appl. Mech. Engrg. 14 (1978), no. 1, 23 – 37. · Zbl 0404.73067 · doi:10.1016/0045-7825(78)90011-7
[15] Roger Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France 96 (1968), 115 – 152 (French). · Zbl 0181.18903
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