Weak Galerkin Finite Element Methods For Second-order Elliptic Equation And Transport-reaction Equation

Since the finite element method was proposed in the 1960s,it has developed sev-eral branches,including conforming finite element method,non-conforming finite ele-ment method,hybrid finite element method,mixed finite element method,non-conforming mixed finite element method,mixed Hybrid finite element method,discontinuous Galerkin finite element method.Based on the discontinuous Galerkin method,Wang and Ye proposed the weak Galerkin finite element method in 2011.Weak Galerkin(WG)method was first developed for numerical solution of second order elliptic equations,in which the classical gradient operator is replaced by the discrete weak gradient and the classical divergence op-erator is replaced by the discrete weak gradient.In recent years,the weak Galerkin finite element method has been widely used,successively used to solve problems such as elliptic equations,biharmonic equations,and transmission equations.This paper briefly describes the development of DG finite element method and weak Galerkin finite element method,introduces the weak Galerkin finite element method of elliptic equation and transport-reaction equation,and introduces weak Galerkin finite ele-ment method without stabilizer.In this paper,a new weak Galerkin finite element method is introduced to solve a class of nonlinear problems.And we proved the convergence of the method,and derived an error estimation formula with the order of O(h~k).Finally,we gave Numerical experiments of weak Galerkin finite element method for solving second-order elliptic equations and transport-reaction equation.