Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A negative-norm least squares method
for Reissner-Mindlin plates

Authors: James H. Bramble and Tong Sun
Journal: Math. Comp. 67 (1998), 901-916
MSC (1991): Primary 65N30, 73V05; Secondary 65F10
MathSciNet review: 1474648
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper a least squares method, using the minus one norm developed by Bramble, Lazarov, and Pasciak, is introduced to approximate the solution of the Reissner-Mindlin plate problem with small parameter $t$, the thickness of the plate. The reformulation of Brezzi and Fortin is employed to prevent locking. Taking advantage of the least squares approach, we use only continuous finite elements for all the unknowns. In particular, we may use continuous linear finite elements. The difficulty of satisfying the inf-sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form, which is very cheap computationally. It is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter $t$. Apart from the simplicity of the elements, the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system. For each diagonal block, one only needs a preconditioner for a second order elliptic problem.

References [Enhancements On Off] (What's this?)

  • 1. D. Arnold and R. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), pp. 1276-1290. MR 91c:65068
  • 2. D. Arnold and R. Falk, Analysis of a linear-linear finite element for the Reissner-Mindlin plate model, Math. Models and Methods in Applied Sciences, 7 (1997), pp. 217-238. MR 98b:73034
  • 3. D. Arnold, R. Falk and R. Winther, Preconditioning in $H(div)$ and applications, Math. Comp. 66 (1997), pp. 957-984. MR 97i:65177
  • 4. D. Arnold, R. Falk and R. Winther, Preconditioning discrete approximations of the Reissner-Mindlin plate model, RAIRO Modél. Math. Anal. Numér. 31 (1997), pp. 517-557. CMP 97:15
  • 5. J. Bramble, Multigrid Methods, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, London. Copublished with John Wiley & Sons, Inc., New York, 1993. MR 95b:65002
  • 6. J. Bramble, R. Lazarov and J. Pasciak, A least-square approach based on a discrete minus one inner product for first order systems, Math. Comp., 66 (1997), pp. 935-955. MR 97m:65202
  • 7. J. Bramble and J. Pasciak, Least-squares methods for Stokes equations based on a discrete minus one inner product, J. Comp. Appl. Math., 74 (1996), pp. 155-173. CMP 97:07
  • 8. J. Bramble and J. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49(1987), pp. 311-329. MR 89b:65234
  • 9. S. Brenner, Multigrid methods for parameter dependent problems, RAIRO Modél. Math. Anal. Numér. 30 (1996), pp. 265-297. MR 97c:73076
  • 10. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994. MR 95f:65001
  • 11. F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plate, Math. Comp. 47 (1986), pp. 151-158. MR 87g:73057
  • 12. F. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models and Methods in Applied Sciences, 1 (1991), pp. 125-151. MR 92e:73030
  • 13. R. Duran and E. Liberman, On mixed finite element methods for Reissner-Mindlin plate models, Math. Comp., 58 (1992), pp. 561-573. MR 92f:65135
  • 14. T. Hughes and L. Franca, A stable bilinear element formulation for Reissner-Mindlin plate theory: Uniform convergence of all higher-order spaces, Comp. Meths. Appl. Mech. Engrg., 67 (1988), pp. 223-240. MR 89g:73033
  • 15. E. Onate, F. Zarate and F. Flores, A simple triangular element for thick and thin plate and shell analysis, Int. J. Numer. Method. Engrg., 37 (1994), pp. 2569-2582.
  • 16. P. Peisker, A multigrid method for Reissner-Mindlin plates, Numer. Math., 59 (1991), 511-528. MR 92g:73083
  • 17. V. Girault and P. Raviart, Finite element approximation of the Navier-Stokes equations. Lecture Notes in Math. #749, Springer-Verlag, New York, 1981. MR 83b:65122
  • 18. R. Stenberg, A new finite element formulation for the plate bending problem, Proceedings of the International Conference on Asymptotic Methods for Elastic Structures, Lisbon, 1993. MR 96k:73084

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N30, 73V05, 65F10

Retrieve articles in all journals with MSC (1991): 65N30, 73V05, 65F10

Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77840

Tong Sun
Affiliation: Institute for Scientific Computation, Texas A&M University, College Station, Texas 77840

Keywords: Plate, locking, least squares, finite element
Received by editor(s): February 7, 1997
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society