Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

  • Authors:

  • Zuozheng Zhang;Ziqing Xie;Zhimin Zhang

  • Affiliations:

  • College of Mathematics and Computer Science, Hunan Normal University, Changsha, People's Republic of China 410081;College of Mathematics and Computer Science, Hunan Normal University, Changsha, People's Republic of China 410081;College of Mathematics and Computer Science, Hunan Normal University, Changsha, People's Republic of China 410081

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2009

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Abstract

Some discontinuous Galerkin methods for the linear convection-diffusion equation 驴驴 u驴+bu驴=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67---96, 2007, we identify superconvergence points for the approximations of u or q=u驴. Our results are twofold:1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively.2) For the consistent DG methods whose numerical fluxes at the mesh nodes converge at the rate of O(h p+1), we prove that the leading term of the discretization error for q is proportional to the Legendre polynomial of degree p. Consequently, the approximation of the gradient superconverges at the zeros of the Legendre polynomial of degree p at the rate of O(h p+1).Numerical experiments are presented to illustrate the theoretical findings.