Least-squares finite element approximations for the Reissner-Mindlin plate

✍ Scribed by Zhiqiang Cai; Xiu Ye; Huilong Zhang

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 106 KB
• Volume: 6
• Category: Article
• ISSN: 1070-5325
• DOI: 10.1002/(sici)1099-1506(199909)6:6<479::aid-nla172>3.0.co;2-k

✦ Synopsis

Based on the Helmholtz decomposition of the transverse shear strain, Brezzi and Fortin in [7] introduced a three-stage algorithm for approximating the Reissner-Mindlin plate model with clamped boundary conditions and established uniform error estimates in the plate thickness. The first and third stages involve approximating two simple Poisson equations and the second stage approximates a perturbed Stokes equation. Instead of using the mixed finite element method which is subject to the 'infsup' condition, we consider a least-squares finite element approximation to such a perturbed Stokes equation. By introducing a new independent vector variable and associated div equation, we are able to establish the ellipticity and continuity of the homogeneous least-squares functional in an H 1 product norm appropriately weighted by the thickness. This immediately yields optimal discretization error estimates for finite element spaces in this norm which are uniform in the thickness. We show that the resulting algebraic equations can be uniformly well preconditioned by wellknown techniques in the thickness. The Reissner-Mindlin model with pure traction boundary condition is also studied. Finally, we consider an alternative least-squares formulation for the perturbed Stokes equation by introducing an independent scalar variable.