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Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element method. (English) Zbl 1095.74031

Summary: Reliable and efficient residual-based a posteriori error estimates are established for the stabilised locking-free finite element methods for Reissner-Mindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do depend neither on the mesh-size nor on the plate thickness, and are uniform for a wide range of stabilisation parameter. The error is controlled in norms that are known to converge to zero in a quasi-optimal way. An adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses \(0.1\), \(0.001\) and \(0.001\).

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] Arnold, D.N., Brezzi, F.: Some new elements for the Reissner-Mindlin plate model. In: J.L. Lions, C. Baiocchi (eds.) Boundary value problems for partial differential equations and applications. Masson, 1993, pp. 287-292. · Zbl 0817.73058
[2] Arnold, SIAM J. Numer. Anal., 26, 1276 (1989) · Zbl 0696.73040 · doi:10.1137/0726074
[3] Babuška, SIAM J. Numer. Anal., 15, 736 (1978) · Zbl 0398.65069 · doi:10.1137/0715049
[4] Bathe, Internat. J. Numer. Methods. Engrg., 28, 1787 (1989) · Zbl 0705.73238 · doi:10.1002/nme.1620280806
[5] Boffi, Numer. Math., 75, 405 (1997) · Zbl 0874.65085 · doi:10.1007/s002110050246
[6] Braess, D., Stability of saddle point problems with penalty, M2AN, 6, 731-742 (1996) · Zbl 0860.65054
[7] Braess, D.: Finite Elements. Cambridge University Press, 1997 · Zbl 0894.65054
[8] Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer-Verlag, 1991 · Zbl 0788.73002
[9] Brezzi, Math. Models and Methods in Appl. Sci., 1, 125 (1991) · Zbl 0751.73053 · doi:10.1142/S0218202591000083
[10] Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Texts in Applied Mathematics 15 Springer New York, 1994 · Zbl 0804.65101
[11] Chapelle, Math. Mod. Meth. Appl. Sc., 8, 407 (3) · Zbl 0907.73055 · doi:10.1142/S0218202598000172
[12] Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element method, M2AN, 33, 1187-1202 (1999) · Zbl 0948.65113
[13] Carstensen, C., Residual-Based A Posteriori Error Estimate for a Nonconforming Reissner-Mindlin Plate Finite Element, SIAM J. Numer. Anal., 39, 2034-2044 (2002) · Zbl 1042.74049
[14] Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, Amsterdam, 1978 · Zbl 0383.65058
[15] Clément, P.: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. R-2, 77-84 (1975) · Zbl 0368.65008
[16] Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numerica, 105-158 (1995) · Zbl 0829.65122
[17] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Berlin-Heidelberg-New York-Tokyo: Springer, 1986 · Zbl 0585.65077
[18] Liebermann, E.: A posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate. Math. Comp. (2000)
[19] Lovadina, SIAM J. Numer. Anal., 33, 2457 (1996) · Zbl 0860.73069 · doi:10.1137/S0036142994265061
[20] Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications, Vol. I. Berlin-Heidelberg-New York: Springer, 1972 · Zbl 0223.35039
[21] Schöberl, J.: Multigrid Methods for a class of parameter dependent problems in primal variables. Technical Report 99-03Spezialforschungsbereich F013, 1999, Johannes Kepler University, 4040-Linz, Austria · Zbl 0957.74059
[22] Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, 1996 · Zbl 0853.65108
[23] Verfürth, R., Robust a posteriori error estimators for singularly perturbed reaction-diffusion equations, Numer. Math., 78, 479-493 (1998) · Zbl 0887.65108
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