Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems

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Abstract

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve p+ 1 order of convergence for the solution and its spatial derivative in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+ 1 towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order p+ 3 / 2 towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the (p+ 1) -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the L2-norm at O(hp + 3 / 2) rate. Finally, we prove that the global effectivity index in the L2-norm converge to unity at O(h1 / 2) rate. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)1868-1904
Number of pages37
JournalJournal of Scientific Computing
Volume76
Issue number3
DOIs
StatePublished - Sep 1 2018

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Local Discontinuous Galerkin Method
Convection-diffusion
Discontinuous Galerkin
Galerkin methods
Error Estimates
Norm
Derivative
Projection
Derivatives
Exact Solution
Polynomials
Converge
A Posteriori Error Estimators
A Priori Error Estimates
Polynomial
Optimal Error Estimates
Superconvergence
Auxiliary Variables
A Posteriori Error Estimates
Piecewise Polynomials

Keywords

  • Gauss–Radau projection
  • Local discontinuous Galerkin method
  • Nonlinear convection–diffusion problems
  • Superconvergence
  • a posteriori error estimation

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Engineering(all)
  • Computational Theory and Mathematics

Cite this

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title = "Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems",
abstract = "In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve p+ 1 order of convergence for the solution and its spatial derivative in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+ 1 towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order p+ 3 / 2 towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the (p+ 1) -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the L2-norm at O(hp + 3 / 2) rate. Finally, we prove that the global effectivity index in the L2-norm converge to unity at O(h1 / 2) rate. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.",
keywords = "Gauss–Radau projection, Local discontinuous Galerkin method, Nonlinear convection–diffusion problems, Superconvergence, a posteriori error estimation",
author = "Mahboub Baccouch",
year = "2018",
month = "9",
day = "1",
doi = "10.1007/s10915-018-0687-9",
language = "English (US)",
volume = "76",
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N2 - In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve p+ 1 order of convergence for the solution and its spatial derivative in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+ 1 towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order p+ 3 / 2 towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the (p+ 1) -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the L2-norm at O(hp + 3 / 2) rate. Finally, we prove that the global effectivity index in the L2-norm converge to unity at O(h1 / 2) rate. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.

AB - In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve p+ 1 order of convergence for the solution and its spatial derivative in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+ 1 towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order p+ 3 / 2 towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the (p+ 1) -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the L2-norm at O(hp + 3 / 2) rate. Finally, we prove that the global effectivity index in the L2-norm converge to unity at O(h1 / 2) rate. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.

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