Discontinuous Galerkin Method For The Singularly Perturbed Reaction Diffusion Equation

In recent years, discontinuous Galerkin(DG) methods has been hotly dis-cussed in scientific computing community. Compared with the finite element method, DG methods approximate the accurate solutions by totally discon-tinuous piecewise polynomials. As a result it performs well in high paralleliz-ability, high-order accuracy. flexibility in choosing the degree of freedom and local compactness. Moreover they can simulate the locally sharp oscillation very well.This paper mainly discusses the numerical solutions of the singularly per-turbed reaction-diffusion equations with Dirichlet boundary condition by dis-continuous Galerkin method on Shishikin mesh and grade mesh. Besides, su-perconvergence is analyzed by the numerical examples. Firstly, the background of the singularly perturbed reaction-diffusion problems are introduced. Sec-ondly, the discontinuous Galerkin method and its discrete formulation are pro-posed. In this paper, we introduce the numerical flux of the LDG method(local discontinuous Galerkin method) and prove the existence and uniqueness of the LDG numerical solution. Finally, numerical experiments in one dimension show that the numerical fluxes have the uniform superconvergence order of 2p+1 at nodes and the numerical solution have the uniform convergence order of p+1 in the sense of L2 norm under both Shishkin and grade meshes. For linear polynomials of two dimensions, the numerical fluxes at nodes have the uniform superconvergence order of 2p+1.