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Mathematics of Computation
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
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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.


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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: http://dx.doi.org/10.1090/S0025-5718-98-00972-7
PII: S 0025-5718(98)00972-7
Keywords: Plate, locking, least squares, finite element
Received by editor(s): February 7, 1997
Article copyright: © Copyright 1998 American Mathematical Society