Date of Award

3-21-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

Xiu Ye

Abstract

The finite element methods (FEMs) are popular numerical methods for solving partial differential equations. They are widely used by mathematicians, scientists and engineers to solve real world problems. The superconvergence in the finite element method (FEM) is a phenomenon where the order of convergence of the numerical solution is higher than the order of convergence of the maximum of the finite element error. Improving the convergence rate by applying certain post-processing techniques to the standard finite element solutions is one of the most important and useful areas of superconvergence. Two model problems will be considered in this dissertation. They are second order elliptic equations and Stokes equations. The objective of the dissertation is to use the $L^2$-projection methods to achieve superconvergence of the finite element solutions for these two model equations. For the second order elliptic problems, superconvergence are investigated for conforming, nonconforming finite element methods and discontinuous Galerkin finite element method by using $L^2$-projection methods. A general superconvergence results are also established for the $H(div)$ finite element approximations of the Stokes equations by using global and local $ L^2$-projection methods. Numerical examples are tested to verify and support the theoretical conclusion.

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