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

DOI:
https://doi.org/10.1090/S0025-5718-98-00972-7

MathSciNet review:
1474648

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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 , 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 . 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.

**1.**Douglas N. Arnold and Richard S. Falk,*A uniformly accurate finite element method for the Reissner-Mindlin plate*, SIAM J. Numer. Anal.**26**(1989), no. 6, 1276–1290. MR**1025088**, https://doi.org/10.1137/0726074**2.**Douglas N. Arnold and Richard S. Falk,*Analysis of a linear-linear finite element for the Reissner-Mindlin plate model*, Math. Models Methods Appl. Sci.**7**(1997), no. 2, 217–238. MR**1440607**, https://doi.org/10.1142/S0218202597000141**3.**Douglas N. Arnold, Richard S. Falk, and R. Winther,*Preconditioning in 𝐻(𝑑𝑖𝑣) and applications*, Math. Comp.**66**(1997), no. 219, 957–984. MR**1401938**, https://doi.org/10.1090/S0025-5718-97-00826-0**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.**James H. Bramble,*Multigrid methods*, Pitman Research Notes in Mathematics Series, vol. 294, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR**1247694****6.**James H. Bramble, Raytcho D. Lazarov, and Joseph E. Pasciak,*A least-squares approach based on a discrete minus one inner product for first order systems*, Math. Comp.**66**(1997), no. 219, 935–955. MR**1415797**, https://doi.org/10.1090/S0025-5718-97-00848-X**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.**James H. Bramble and Joseph E. Pasciak,*New convergence estimates for multigrid algorithms*, Math. Comp.**49**(1987), no. 180, 311–329. MR**906174**, https://doi.org/10.1090/S0025-5718-1987-0906174-X**9.**Susanne C. Brenner,*Multigrid methods for parameter dependent problems*, RAIRO Modél. Math. Anal. Numér.**30**(1996), no. 3, 265–297 (English, with English and French summaries). MR**1391708**, https://doi.org/10.1051/m2an/1996300302651**10.**Susanne C. Brenner and L. Ridgway Scott,*The mathematical theory of finite element methods*, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR**1278258****11.**F. Brezzi and M. Fortin,*Numerical approximation of Mindlin-Reissner plates*, Math. Comp.**47**(1986), no. 175, 151–158. MR**842127**, https://doi.org/10.1090/S0025-5718-1986-0842127-7**12.**Franco Brezzi, Michel Fortin, and Rolf Stenberg,*Error analysis of mixed-interpolated elements for Reissner-Mindlin plates*, Math. Models Methods Appl. Sci.**1**(1991), no. 2, 125–151. MR**1115287**, https://doi.org/10.1142/S0218202591000083**13.**Ricardo Durán and Elsa Liberman,*On mixed finite element methods for the Reissner-Mindlin plate model*, Math. Comp.**58**(1992), no. 198, 561–573. MR**1106965**, https://doi.org/10.1090/S0025-5718-1992-1106965-0**14.**Thomas J. R. Hughes and Leopoldo P. Franca,*A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all higher-order spaces*, Comput. Methods Appl. Mech. Engrg.**67**(1988), no. 2, 223–240. MR**929284**, https://doi.org/10.1016/0045-7825(88)90127-2**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.**Petra Peisker,*A multigrid method for Reissner-Mindlin plates*, Numer. Math.**59**(1991), no. 5, 511–528. MR**1121656**, https://doi.org/10.1007/BF01385793**17.**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. MR**548867****18.**Rolf Stenberg,*A new finite element formulation for the plate bending problem*, Asymptotic methods for elastic structures (Lisbon, 1993) de Gruyter, Berlin, 1995, pp. 209–221. MR**1333214**

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Additional Information

**James H. Bramble**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77840

Email:
bramble@math.tamu.edu

**Tong Sun**

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

Email:
tsun@math.tamu.edu

DOI:
https://doi.org/10.1090/S0025-5718-98-00972-7

Keywords:
Plate,
locking,
least squares,
finite element

Received by editor(s):
February 7, 1997

Article copyright:
© Copyright 1998
American Mathematical Society