A superconvergence result for discontinuous Galerkin methods applied to elliptic problems

This paper presents a theoretical and numerical study of a class of discontinuous Galerkin methods that shows the approximation of the gradient superconverges at the zeros of the Legendre polynomials on a model 1D elliptic problem. Numerical experiments validate the theoretical results.

[1]  The validity of the superconvergent patch recovery in discontinuous finite element formulations , 2000 .

[2]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[3]  M. Bakker.  One-dimensional Galerkin methods and superconvergence at interior nodal points , 1984 .

[4]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[5]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .

[6]  PAUL CASTILLO,et al.  Performance of Discontinuous Galerkin Methods for Elliptic PDEs , 2002, SIAM J. Sci. Comput..

[7]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[8]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[9]  M. Bakker.  A note onCo Galerkin methods for two-point boundary problems , 1982 .

[10]  Ilaria Perugia,et al.  Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids , 2001, SIAM J. Numer. Anal..

[11]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[12]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[13]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[14]  Paul Castillo,et al.  An Optimal Estimate for the Local Discontinuous Galerkin Method , 2000 .