### 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 language | English (US) |
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Pages (from-to) | 1130-1153 |

Number of pages | 24 |

Journal | Computers and Mathematics with Applications |

Volume | 67 |

Issue number | 5 |

DOIs | |

State | Published - Mar 1 2014 |

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### 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

**Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension

AU - Baccouch, Mahboub

PY - 2014/3/1

Y1 - 2014/3/1

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.

KW - Local discontinuous Galerkin method

KW - Radau points

KW - Superconvergence

KW - Transient convection-diffusion problems

KW - a posteriori error estimation

UR - http://www.scopus.com/inward/record.url?scp=84894581709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894581709&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2013.12.014

DO - 10.1016/j.camwa.2013.12.014

M3 - Article

VL - 67

SP - 1130

EP - 1153

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 5

ER -