Global convergence of a posteriori error estimates for the discontinuous Galerkin method for one-dimensional linear hyperbolic problems

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Abstract

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

Original languageEnglish (US)
Pages (from-to)172-193
Number of pages22
JournalInternational Journal of Numerical Analysis and Modeling
Volume11
Issue number1
StatePublished - Jan 1 2014

Fingerprint

Hyperbolic Problems
A Posteriori Error Estimates
Discontinuous Galerkin Method
Order of Convergence
Galerkin methods
Global Convergence
Mesh Refinement
Converge
Norm
Superconvergence
Piecewise Polynomials
Smooth Solution
Error Estimates
Numerical Simulation
Polynomials
Computer simulation

Keywords

  • Discontinuous Galerkin method
  • Hyperbolic problems
  • Residual-based a posteriori error estimates
  • Superconvergence

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

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abstract = "In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.",
keywords = "Discontinuous Galerkin method, Hyperbolic problems, Residual-based a posteriori error estimates, Superconvergence",
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N2 - In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

AB - In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

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