Domain Decomposition Method Of Mixed Finite Element

Domain decomposition method is one of the most effective numeric methods on solving partial differential equations. It is generally used with normative finite element method. In this paper, we mainly consider applying it to non-normative finite element method which is called mixed finite element method. The paper contains five chapters. In chapter 1, we first introduce finite element and the steps how to apply it to solve partial differential equation, then bring in mixed finite element and domain decomposition method, finally, we rise the main problem what we want to solve in the paper. In chapter 2, we introduce the basic theories about mixed finite element. We introduce the existence and uniqueness theorem of mixed finite element, and then show mixed finite element formula about some equations and the conclusion whether the solution of the formula is existence and uniqueness. In chapter 3, we introduce domain decomposition method. We introduce superposition and non-superposition domain decomposition method apart. About superposition domain decomposition method, based on Lions framework, we introduce Schwarz alternating method, about non-superposition domain decomposition method, based on Steklov-Poincare operator, we introduce D-N alternating method. In chapter 4, we introduce domain decomposition method of mixed finite element method. We introduce a method which based on Schwarz alternating method, we show the computing formula and relative discrete finite element, and based on Lions framework and extremum principle respectively, we prove its convergence in different norms. In chapter 5, we make a brief summarize and what we will do in next phase. In this paper, it is showed that domain decomposition method can also be used with mixed finite element.