Date of Award

6-21-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

Xiu Ye

Abstract

A large number of partial differential equations in geoscience and engineering can be described as interface problems. Three interface models will be considered in this dissertation. They are elliptic interface problems, Stokes interface problems, and Brinkman problems. The purpose of the dissertation is to develop and analyze robust finite element methods and finite volume methods for solving these problems. Firstly, we report the a posteriori error estimate for interface problems by conforming finite volume methods, nonconforming finite volume methods, and discontinuous finite volume methods. Recovery-based and residual-based a posteriori error estimators are constructed by using computable quantities. Secondly, we develop weak Galerkin finite element methods for solving these three kinds of interface problems. The weak Galerkin finite element method is a new numerical method for solving partial differential equations. It uses weakly defined differential operators to provide desirable flexibilities. For interface problems, such a flexibility gives rise to robust numerical schemes. For all this work, numerical examples are investigated to support the theoretical conclusion.

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