Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension

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Abstract

In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver-gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to (p + l)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p+2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to (p+l)-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at O(hp+2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. 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 p + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)511-531
Number of pages21
JournalJournal of Computational Mathematics
Volume34
Issue number5
DOIs
StatePublished - Sep 1 2016

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Local Discontinuous Galerkin Method
Optimal Error Estimates
Convection-diffusion Problems
A Posteriori Error Estimates
Galerkin methods
Discontinuous Galerkin
Polynomials
Order of Convergence
Converge
Norm
Polynomial
Derivative
Mesh Refinement
Piecewise Polynomials
Derivatives
Smooth Solution
Error term
Error Estimates
Directly proportional
Numerical Experiment

Keywords

  • A posteriori error estimation
  • Convection-diffusion problems
  • Local discontinuous Galerkin method
  • Radau polynomials
  • Super-convergence

ASJC Scopus subject areas

  • Computational Mathematics

Cite this

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title = "Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension",
abstract = "In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver-gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to (p + l)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p+2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to (p+l)-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at O(hp+2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. 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 p + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.",
keywords = "A posteriori error estimation, Convection-diffusion problems, Local discontinuous Galerkin method, Radau polynomials, Super-convergence",
author = "Mahboub Baccouch",
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T1 - Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension

AU - Baccouch, Mahboub

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N2 - In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver-gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to (p + l)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p+2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to (p+l)-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at O(hp+2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. 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 p + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.

AB - In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver-gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to (p + l)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p+2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to (p+l)-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at O(hp+2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. 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 p + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.

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