The Uzawa-Type Finite Element Methods For Nearly Incompressible Elasticity Problems

In this thesis, the research focuses on the Uzawa-type finite element methods which could overcome the locking problem of nearly incompressible elasticity problems. When the elastic material becomes incompressible, i.e.,,when the Lame coefficient ??+?,it is well known that phenomenon of locking occurs if we use the normal lower order finite elements, that is to say, the convergence rate of finite element approximations deteriorates. For the nearly incompressible isotropic elasticity problems, by bringing in an extra "pressure" variable, we can translate the elasticity problems into a saddle-point systems and by coupling the classical Uzawa-type methods which could convert the saddle-point systems to elliptic problems with mixed finite elements and multigrid methods and adaptive finite element methods we develop three effective iteration methods, we present the convergent results and provability of the algorithms. With discussion in the thesis we can find out that these three methods are locking-free for nearly incompressible elasticity problems when we use lower order finite elements and the finite element spaces don't have to satisfy discrete LBB condition. Some numerical examples are presented to confirm the results in the end.