Domain Decomposition Methods for Reissner - Mindlin Plates discretized with the Falk -

The Reissner-Mindlin plate theory models a thin plate with thickness t. The condition number of finite element approximations of this model deteriorates badly as the thickness t of the plate converges to 0. In this thesis, we develop an overlapping domain decomposition method for the Reissner-Mindlin plate model discretized by Falk-Tu elements with a convergence rate which does not deteriorate when t converges to 0. We use modern overlapping methods which use the Schur complements to define coarse basis functions and show that the condition number of this overlapping method is bounded by C(1 + H δ )(1 + log h ). Here H is the maximum diameter of the subdomains, δ the size of overlap between subdomains, and h the element size. Numerical examples are provided to confirm the theory. We also modify the overlapping method to develop a BDDC method for the Reissner-Mindlin model. We establish numerically an extension lemma to obtain a constant bound and an edge lemma to obtain a C(1+ log h ) bound. Given such bounds, the condition number of this BDDC method is shown to be bounded by C(1 + log h ).