SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN AND LOCAL DISCONTINUOUS GALERKIN SCHEMES FOR LINEAR HYPERBOLIC AND CONVECTION-DIFFUSION EQUATIONS IN ONE SPACE DIMENSION

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 equations. We 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 equations. The order of superconvergence for both cases is proved to be k + 3/2 when piecewise P-k polynomials with k >= 1 are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise Pk polynomials with arbitrary k >= 1, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise P-1 polynomials.