Stability and analysis for finite element approximations of elliptic problems

Abstract

In this thesis, I am concerned with two topics on Finite Element Method of elliptic problems : Construction of stable finite element in three dimensional space and a posteriori error estimate for the mixed element method. In Chapter 2, the stable finite element approximation scheme for the Stokes problem in three dimensional space will be analyzed. We will prove the stability condition for some higher degree finite spaces through the modified macroelement condition. Thus, We will show that the mixed finite element method with pressure stabilization (2.2) converges for some higer order conforming tetrahedral elements. In Chapter 3, we will propose a error estimator for mixed element method of second order ellipitc problems. We will present an error estimator for mixed element method. This estimator will be only computed by edges and be proved to be efficient in sense that it is global upper bounded to the true error. We will present the mixed discretization and a postprocessing technique. whic is based on the elimination of the continuity constraints for the normal components of the flux on the interelement boundaries from the Raviart-Thomas space.