The Weak Galerkin Finite Element Method For A Class Of Stochastic Partial Differential Equations

The weak Galerkin finite element method is a class of recently and rapidly develope-d numerical tools for approximating partial differential equations.Unlike the standard Galerkin method,its trial and test function spaces consist of totally discontinuous piece-wisely defined polynomials in the whole domain.This method has been vastly applied to many fields[29,35,38,54,61-63].In this paper,we will apply this method to solve a class of linear stochastic parabolic partial differential equations with additive noise,we set up a semi-discrete and a fully-discrete numerical scheme,respectively,for the linear stochastic partial differential equations and derive the optimal order for error estimates in the sense of strong convergence.This paper is divided into six chapters,the chapter 1 concludes the application background and meaning of the problems we study in this paper;in the chapter 2 we introduce some basic knowledge about the stochastic problems,and give the definitions of the Ito integration in infinite dimension spaces,the linear stochastic partial differential equations with additive noise and their mild solutions;in the chapter 3 we introduce the basic ideas of the weak Galerkin finite element method,include the RT elements weak Galerkin finite element method and the weak Galerkin element method with stability term;in the chapter 4 we study the linear stochastic partial differential equations with additive noise by the weak Galerkin method with RT elements and get the optimal strong convergence error estimate;in the chapter 5 we study the linear stochastic partial differential equations with additive noise by the weak Galerkin method with stability term and also get the optimal strong convergence error estimate;in the chapter 6 we conclude this paper and discuss the further plan of study.