Preconditioning for the mixed formulation of linear plane elasticity

by Wang, Yanqiu

In this dissertation, we study the mixed ?nite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed ?nite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The ?nite element discretization of the mixed formulation leads to a symmetric inde?nite linear system. Next, we study e?cient iterative solvers for the symmetric inde?nite linear system which arises from the mixed ?nite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the inde?nite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther ?nite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.