Weak Galerkin Finite Element Method To Solve Two Kinds Of Evolution Equations

The numerical method of two kinds of evolution equations in realistic question were mainly discussed in this paper. Based on systematically study the basic of finite element theory, the weak Galerkin finite element method and the Stokes projection were combined to solve the parabolic Naive-Stokes questions and Stokes differential-integral equations.Different from the standard Galerkin finite element method, weak Galerkin finite element method introduce a weak gradient operator to replace the traditional gradient operator,and introduce stable operator to realize the continuity of numerical solution in the form of a weak solution. Introduction of Stokes projection is greatly reduced the difficulty of solving.In this paper, the difficulty of numerical method for solving parabolic Naiver-Stokes and Stokes differential-integral equations were reduced by weak Galerkin finite element method. Meanwhile, the validity and stability of the weak Galerkin finite element method are proved by studying the existence, uniqueness and convergence of solution for above equation, further illustrating the superiority of the weak Galerkin finite element method.This paper is composed of four chapters as follows.In chapter one, we introduce the research background and the application prospect of this paper. Meanwhile, we briefly introduce the partial differential equations, the finite element method, the weak Galerkin finite element method and the evolution equations.In chapter two, we introduce the preparatory knowledge we need in this paper,such as the properties of Sobolev space, related inequalities, and the finite element space.In chapter three, we show that the Naiver-Stokes equation are solved by weak Galerkin finite element method. Firstly, we construct the semi-discrete weak Galerkinfinite scheme to prove the existence and uniqueness of solution. Then the error estimates are obtained by the introduction of Stokes projection approximation.In chapter four, we show that the Stokes type differential-integral equation are solved by weak Galerkin finite element method. we also construct the weak Galerkin finite scheme to prove the existence and uniqueness of solution. The error estimates are obtained by the introduction of Stokes projection approximation.