Superconvergence And Uniform Convergence Of DG Methods For Convection-Diffusion Equations

The discontinuous Galerkin method(DGM) was first introduced for the neutron transport equation(The first-order hyperbolic equation) in 1973 by Reed and Hill[72].Since then there has been an active development of DG methods for hyperbolic and elliptic equations in parallel.In 1997,Bassi and Rebay[11]provided a discontinuous Galerkin method for solving compressible Navier-Stokes equations,and obtained a stable and high-order convergent scheme.Motivated by the successful numerical experiments of Bassi and Rebay, Cockburn and Shu[36]developed the local discontinuous Galerkin method (LDG).At the same time,a discontinuous Galerkin method was introduced by Baumann and Oden[8].Now the DG methods have been used widely for hyperbolic conservation laws systems,elliptic equations,convection-diffusion equations,Hamilton-Jacobi equations and KdV equations,etc.For a fairly thorough compilation of the history of these methods and their applications see[31].In recent years,the DG methods for convection-diffusion problems have been one of the highlights in the study of numerical methods.Inspired by the great success of the DG method in solving hyperbolic equations,in this paper the discontinuous Galerkin method for convection-diffusion equation would be studied.The existence and uniqueness of the approximate solutions for a class of DG methods are proved.Then we will present the asymptotic expansions of the discretization errors for both the potential u and its derivative q = u' for a class of DG methods.For md-LDG method,the leading terms for the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p + 1 on each element,respectively.This fact implies that the p + 1 degree right Radau points and left Radau points are the p + 2 degree superconvergence points for U and Q,separately.Nevertheless, for other DG methods which are consistent and conservative and satisfy some assumptions,the leading terms for the errors of U and Q are proportional to Legendre polynomial of degree p,respectively.As a result,only the p+1-degree Gauss points are the p + 1-order superconvergence points for Q.A Posteriori Error estimates based on superconvergence property will be developed.Our numerical experiment verify our theoretical results.On the other hand,whenεis small,under the uniform mesh,the classical numerical methods cannot produce a scheme of uniform convergence.In this paper we will compare two-type layer-adapted meshes,i.e.,Shishkin mesh and improved grade meshes,when they are used in the h-version of the LDG method for one and two dimensional problems.The numerical results exhibit that the LDG method does not produce any oscillation even under uniform meshes for arbitraryεfor both 1-D and 2-D cases.On the other hand,the 2p + 1-order uniform superconvergence of numerical fluxes are observed numerically for the LDG method under both the Shishkin and improved grade meshes.The numerical results indicate that the improved grade meshes not only keep the advantages of the Shishkin meshes,but is also more efficient and stable than the Shishkin meshes.It is worthwhile to point out that theoretical analysis of the uniform convergence is extremely difficult and remains an open problem for the LDG methodA robust DG scheme will be designed to solve the singularly perturbed convection-diffusion equations in one-dimensional setting with Dirichlet boundary conditions.The existence and uniqueness of approximate solutions are proved.The 2p + 1-order superconvergence of numerical traces at each node is observed.We also note that this DG method is robust with respect to the diffusion coefficientεunder the improved grade mesh.