Robust BDDC Preconditioners for Reissner–Mindlin Plate Bending Problems and MITC Elements

Abstract

A Balancing Domain Decomposition Method by Constraints (BDDC) is constructed and analyzed for the Reissner–Mindlin plate bending problem discretized with Mixed Interpolation of Tensorial Components (MITC) finite elements. This BDDC algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints. After the implicit elimination of the interior degrees of freedom in each subdomain, the resulting plate Schur complement is solved by the preconditioned conjugate gradient method. The preconditioner is based on the solution of local Reissner–Mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse Reissner–Mindlin plate problem for the primal degrees of freedom. The main results of the paper are the proof and numerical verification that the proposed BDDC plate algorithm is scalable, quasi-optimal, and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem, both the interface plate problem and the preconditioner are positive definite. The numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties.