Finite element approximations of elliptic problems

Abstract

In this thesis, I am concerned with finite element approximation schemes of elliptic boundary value problems. Specially, least-squares mixed methods for second order elliptic problems, stabilization procedure for Stokes problem using nonconforming quadrilateral finite element and a parallel iterative Galerkin scheme for second order elliptic problems will be studied. In Chapter 1, a theoretical analysis of a least-squares mixed finite element method for second-order elliptic problems having non-symmetric matrix of coefficients is presented. It is proved that the least-squares mixed method does not subject to the Ladyzhenskaya-Babuska-Brezzi[25] (LBB) consistency condition, and that the finite element approximation yields a symmetric positive definite linear system with condition number $O(h^{-2})$. Optimal order error estimates are developed. In Chapter 2, stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by[20] added by conforming bubbles to the velocity and discontinuous piecewise linear to the pressure on quadrilateral elements. Optimal order error estimates are derived. In Chapter 3, a parallel iterative Galerkin method based on domain decomposition technique with nonconforming quadrilateral finite elements will be analyzed for second-order elliptic equations subject to the Robin boundary condition. Optimal order error estimates are derived with respect to a broken $H^1$ norm and $L^2$ norm. Applications to time-dependent problems will be considered. Some numerical experiments supporting the theoretical results will be given. This chapter is to extend the work in [5] to the non-selfadjoint case of second order equations including the term ▽u. We suppose that uniformly ellipticity holds. Hence the arguments in [5] may be applied, word for word.