Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension

Mahboub Baccouch, Helmi Temimi

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L2-norm, respectively, when p-degree piecewise polynomials with p≥1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.

Original languageEnglish (US)
Pages (from-to)403-434
Number of pages32
JournalInternational Journal of Numerical Analysis and Modeling
Volume13
Issue number3
StatePublished - Jan 1 2016

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Optimal Error Estimates
Convection-diffusion Problems
Discontinuous Galerkin
Superconvergence
Discontinuous Galerkin Method
Galerkin methods
Derivatives
Shishkin Mesh
Derivative
Optimal Rate of Convergence
Piecewise Polynomials
Convergence Rate
Boundary Layer
Boundary layers
Numerical Experiment
Polynomials
Oscillation
Norm
Demonstrate
Experiments

Keywords

  • Convection-diffusion problems
  • Discontinuous Galerkin method
  • Shishkin meshes
  • Singularly perturbed problems
  • Superconvergence
  • Upwind and downwind points

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

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abstract = "In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L2-norm, respectively, when p-degree piecewise polynomials with p≥1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.",
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N2 - In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L2-norm, respectively, when p-degree piecewise polynomials with p≥1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.

AB - In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L2-norm, respectively, when p-degree piecewise polynomials with p≥1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.

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