Preconditioning for the mixed formulation of linear plane elasticity
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In this dissertation, we study the mixed ﬁnite 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 ﬁnite 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 ﬁnite element discretization of the mixed formulation leads to a symmetric indeﬁnite linear system. Next, we study eﬃcient iterative solvers for the symmetric indeﬁnite linear system which arises from the mixed ﬁnite 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 indeﬁnite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther ﬁnite 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.
Mixed finite element method
Overlapping Schwarz method
Wang, Yanqiu (2004). Preconditioning for the mixed formulation of linear plane elasticity. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from