Peisker, P.; Braess, Dietrich Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates. (English) Zbl 0758.73050 RAIRO, Modélisation Math. Anal. Numér. 26, No. 5, 557-574 (1992). Summary: The mixed-interpolated elements of K.-J. Bathe and E. N. Dvorkin [Int. J. Numer. Methods Eng. 21, 367-383 (1985; Zbl 0551.73072)] and K.-J. Bathe, F. Brezzi and S. W. Cho [Comput. Struct. 32, No. 3/4, 797-814 (1989; Zbl 0705.73241)] are analyzed. It is shown that convergence is uniform in the thickness parameter when the Mindlin- Reissner plate is treated. To this end a discrete analog of the Helmholtz decomposition of \(L_ 2\) is introduced. Cited in 17 Documents MSC: 74S05 Finite element methods applied to problems in solid mechanics 74K20 Plates Keywords:thickness parameter; Helmholtz decomposition Citations:Zbl 0551.73072; Zbl 0705.73241 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] D. N. ARNOLD & R. S. FALK, A uniformly accurate finite element method for the Mindlin-Reissner plate, SIAM J. Numer. Anal. 26, 1276-1290 (1989). Zbl0696.73040 MR1025088 · Zbl 0696.73040 · doi:10.1137/0726074 [2] K. J. BATHE & F. BREZZI, On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. In Proceeding Conf. on Mathematics of Finite Elements and Applications 5 (J. R. Whiteman, ed.), Academic Press 1985, pp. 491-503. Zbl0589.73068 MR811058 · Zbl 0589.73068 [3] K. J. BATHE, F. BREZZI & S. W. CHO, The MITC7 and MITC9 plate bending elements, J. Computers & Structures 32, 797-814 (1989). Zbl0705.73241 · Zbl 0705.73241 · doi:10.1016/0045-7949(89)90365-9 [4] K. J. BATHE & E. N. DVORKIN, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, Int. J. Num. Meth. Eng. 21, 367-383 (1985). Zbl0551.73072 · Zbl 0551.73072 · doi:10.1002/nme.1620210213 [5] F. BREZZI, K. J. BATHE & M. FORTIN, Mixed-interpolated elements for Reissner-Mindlin plates, Int J. Num. Meth. Eng. 28, 1787-1801 (1989). Zbl0705.73238 MR1008138 · Zbl 0705.73238 · doi:10.1002/nme.1620280806 [6] [6] F. BREZZI, J. Jr. DOUGLAS, M FORTIN & L. D. MARINI, Efficient rectangular mixed finite elements in two and three space variables. RAIRO Model. Math. Anal. Numér. 21, 581-604 (1987). Zbl0689.65065 MR921828 · Zbl 0689.65065 [7] [7] F. BREZZI, J. Jr. DOUGLAS, & L. D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47, 217-235 (1985). Zbl0599.65072 MR799685 · Zbl 0599.65072 · doi:10.1007/BF01389710 [8] F. BREZZI & M. FORTIN, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47, 151-158 (1986). Zbl0596.73058 MR842127 · Zbl 0596.73058 · doi:10.2307/2008086 [9] F. BREZZI & J. PITKÄRANTA, On the stabilization of finite element approximations of the Stokes equations. In < Efficient Solutions of Elliptic Systems > (W. Hackbusch, ed.), Vieweg, Braunschweig 1984, 11-19. Zbl0552.76002 MR804083 · Zbl 0552.76002 [10] P. G. CIARLET, The Finite Element Method for Elliptic Problems. North Holland 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058 [11] J. Jr. DOUGLAS & J. E. ROBERTS, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44, 39-52 (1985). Zbl0624.65109 MR771029 · Zbl 0624.65109 · doi:10.2307/2007791 [12] L. P. FRANCA & R. STENBERG, A modification of a low order Reissner-Mindlin plate bending element. Rapport de Recherche No 1084 INRIA-Rocquencourt, France, 1989. · Zbl 0749.73076 [13] V. GIRAULT & P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. Zbl0585.65077 MR851383 · Zbl 0585.65077 [14] [14] Z. HUANG, A multi-grid algorithm for mixed problems with penalty, Math. 57, 227-247 (1990). Zbl0712.73106 MR1057122 · Zbl 0712.73106 · doi:10.1007/BF01386408 [15] P. A. RAVIART & J. M. THOMAS, A mixed finite element method for second order elliptic problems In < Mathematical Aspects of Finite Element Methods > (I. Galligani and E. Magenes, eds.), pp 292-315, Springer Verlag 1977. Zbl0362.65089 MR483555 · Zbl 0362.65089 [16] [16] L. R. SCOTT & M. VOGELIUS, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Model. Math. Anal. Numér. 19, 111-143 (1985). Zbl0608.65013 MR813691 · Zbl 0608.65013 [17] F. BREZZI & M. FORTIN, Mixed and Hybrid Finite Elements, Springer Verlag 1991. MR1115205 · Zbl 0788.73002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.