Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation

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Abstract

In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L<sup>2</sup> error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L<sup>2</sup>-norm under mesh refinement. The order of convergence is proved to be p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L<sup>2</sup>-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)1461-1491
Number of pages31
JournalNumerical Methods for Partial Differential Equations
Volume31
Issue number5
DOIs
StatePublished - Sep 1 2015

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Discontinuous Galerkin
Wave equations
Second Order Equations
Local Discontinuous Galerkin Method
Error Estimates
Wave equation
Converge
Norm
Derivative
A Posteriori Error Estimation
Optimal Error Estimates
Superconvergence
Mesh Refinement
A Posteriori Error Estimates
Piecewise Polynomials
Order of Convergence
Smooth Solution
Galerkin methods
Convergence Rate
Computational Results

Keywords

  • local discontinuous Galerkin method
  • residual-based a posteriori error estimates
  • second-order wave equation
  • superconvergence

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics
  • Analysis

Cite this

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abstract = "In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.",
keywords = "local discontinuous Galerkin method, residual-based a posteriori error estimates, second-order wave equation, superconvergence",
author = "Mahboub Baccouch",
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language = "English (US)",
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N2 - In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

AB - In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

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