Date of Award

6-2-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

Xiu Ye

Abstract

Mixed boundary conditions are practical situations that appear in most potential and physics problems. In this dissertation, a weak Galerkin (WG) finite element method is proposed and analyzed for the general elliptic equation with mixed boundary conditions. Furthermore, we developed a modified weak Galerkin (MWG) method for the second order elliptic problem with mixed boundary conditions. While keeping the same accuracy, the modified weak Galerkin method reduces the degree of freedom of the original weak Galerkin method by eliminating unknowns associated with element boundaries. Optimal order error estimates are established in both a discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG and MWG approximations. The numerical experiments are presented to verify the efficiency of the method.

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