Overlapping Domain Decomposition Methods for Numerically Thin Reissner--Mindlin Plates Approximated with the Falk--Tu Elements

Abstract

The Reissner--Mindlin plate theory models a thin plate with thickness $t$. The condition numbers of finite element approximations of this model deteriorate badly as $t$ approaches $0.$ An overlapping domain decomposition method for the Reissner--Mindlin plate model discretized by the Falk--Tu elements is developed. A modern overlapping method which uses the subdomain Schur complements to define coarse basis functions is used and it is shown that the condition number of this overlapping method is bounded by $C(1+\frac{H}{\delta})^3(1+\log\frac{H}{h})^2$ if $t$ is smaller than or comparable to the element size $h$. Here $H$ is the maximum diameter of the subdomains and $\delta$ is the size of overlap between the subdomains. Numerical examples which confirm the theory are provided.