A Negative - Norm Least Squares Method for Plates

In this paper a least squares method, using the minus one norm developed in 6] and 7], 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 nite elements for all the unknowns. In particular, we may use continuous linear nite elements. The diiculty 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. 1. Introduction The numerical solution of the Reissner-Mindlin plate model has been discussed by many authors. Many schemes developed early on, using standard nite element methods, are known to possess a locking problem for very thin plates. Using a Helmholtz decomposition, Brezzi and Fortin 11] derived a reformulation of the Reissner-Mindlin plate model, which fundamentally removed the reason for the locking. Another advantage of this reformulation is that the four unknowns are decoupled into two elliptic problems and one mixed saddle point problem. Brezzi and Fortin gave a scheme based on this for which they proved error estimates