A Negative Norm Least-Squares Method for Reissner-Mindlin Plates

Tong Sun

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


A negative norm least squares method is introduced to approximate the solution of the Reissner-Mindlin plate problems 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 proven 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.