Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension

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Abstract

In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+3) and O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2-norm converge to unity at O(h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.

Original languageEnglish (US)
Pages (from-to)1130-1153
Number of pages24
JournalComputers and Mathematics with Applications
Volume67
Issue number5
DOIs
StatePublished - Mar 1 2014

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Local Discontinuous Galerkin Method
Convection-diffusion Problems
Superconvergence
A Posteriori Error Estimates
Galerkin methods
Discontinuous Galerkin
Error Estimates
Derivative
Discretization Error
Derivatives
Polynomial
Polynomials
Converge
A Posteriori Error Estimation
Boundary conditions
Neumann Boundary Conditions
Numerical Computation
Dirichlet Boundary Conditions
Dirichlet
Directly proportional

Keywords

  • Local discontinuous Galerkin method
  • Radau points
  • Superconvergence
  • Transient convection-diffusion problems
  • a posteriori error estimation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

Cite this

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title = "Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension",
abstract = "In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+3) and O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2-norm converge to unity at O(h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.",
keywords = "Local discontinuous Galerkin method, Radau points, Superconvergence, Transient convection-diffusion problems, a posteriori error estimation",
author = "Mahboub Baccouch",
year = "2014",
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doi = "10.1016/j.camwa.2013.12.014",
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volume = "67",
pages = "1130--1153",
journal = "Computers and Mathematics with Applications",
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AU - Baccouch, Mahboub

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N2 - In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+3) and O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2-norm converge to unity at O(h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.

AB - In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+3) and O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2-norm converge to unity at O(h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.

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