Weak Galerkin Finite Element Method For The Partial Integro-differential Equation With Weakly Singular Kernel

Weak Galerkin finite element method for partial integro-differ-ential equation with weakly singular kernel is considered in the pa-per.The weak Galerkin finite element method is used for the space discretization,and the backward Euler approach is employed for the time discretization.The integral term is approximated by the piecewise constant function.Then we can derive the fully discrete weak Galerkin finite element scheme.The stability and error es-timate for the fully discrete scheme will be given.And numerical experiments verify ourt theory findings.The weak Galerkin finite element method was first presented by Wang to solve the second order elliptic problems.Since then,some modified weak Galerkin methods have also been studied,such as,elliptic problem,Helmholtz equation,Stokes equation,Brinkman equation.In comparison to standard Galerkin finite el-ement method,the weak Galerkin finite element method provides the discrete weak function and weak derivative,and allows the use of totally discontinuous piecewise polynomials in the finite element procedure.And the endpoint values may not be necessarily the internal values.So the weak Galerkin finite element method have an advantage of discontinuous finite element method.This article is organized as follows:Chapter 1 presents the current research status at home and aborad as well as the signif-icance of the partial integro-differential equation.In Chapter 2,we will give the preliminary knowledge for the research.Chapter 3 describes the fully discrete weak Galerkin finite element scheme for problem.The stability and error estimate for the fully discrete scheme are derived in Chapter 4.In Chapter 5,numerical exper-iments are provided to illustrate the weak Galerkin finite element method is reliable and effective.This paper ends with a brief con-clusion and prospect for the follow-up research.