Superconvergence Of The LDG Method For Convection-Diffusion Equations In One Dimension

Now, more and more people begin to pay their attention to the development and research of superconvergence property, because they can obtain a more accurate numerical solution than classical convergent equations, and the error is stable in a certain range. Convection-diffusion equations are a kind of basic mathematical physical equations, they can describe the transport of many kinds of energy in fluid flow as well as some chemical reactions and other physical phenomena. As everyone knows, the standard Galerkin finite element scheme is not stable, unless meshes step size is sufficient small, but this selection will result in more difficult to calculate and larger amount of calculation.In this paper, we study the superconvergence property for the discontinuous Galerkin(DG) and the local discontinuous Galerkin(LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equtions. Both analysises are based on the first boundary conditions. Specifically, wo prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equtions. The order of superconvergence for both cases is proved to be κ+3/2when piecewise Pkpolynomials with κ≥1 are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise Pk polynomials with arbitrary κ≥ 1. The proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise Pl　polynomials, this paper will improve upon that results.