The Weak Galerkin Methods For The Reaction-advection-diffusion Equations

In this paper,we consider the weak Galerkin(WG)finite element method(FEM)for solving the second-order elliptic problems with the first-order term under several boundary conditions.The WG methods are a new class of finite element discretizations for solving partial differential equations(PDE).The key of the WG methods is that we enforce the continuity weakly.In the WG method,the classical differential operators are replaced by the generalized differential operators and we use stabilizers to keep the algorithm stable.The WG methods show the advantages in several aspects.For example,it is more convenient to construct the high-order approximation spaces than the traditional FEM methods.And the WG method is easy to be used on polygonal meshes thanks to the weakly requirement of the continuity.The second-order elliptic problems with a first-order term are solved in this paper,and we shall show how the WG method work on the problem under the Dirichlet,Neumann,and Robin boundary conditions.The error equations are given,and the optimal convergence orders are achieved.