### 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 L^{2}-norm at O(h^{p+2}) rate. Finally, we prove that the global effectivity indices in the L^{2}-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 P^{p} polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.

Original language | English (US) |
---|---|

Pages (from-to) | 511-531 |

Number of pages | 21 |

Journal | Journal of Computational Mathematics |

Volume | 34 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

**Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension

AU - Baccouch, Mahboub

PY - 2016/9/1

Y1 - 2016/9/1

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.

KW - A posteriori error estimation

KW - Convection-diffusion problems

KW - Local discontinuous Galerkin method

KW - Radau polynomials

KW - Super-convergence

UR - http://www.scopus.com/inward/record.url?scp=85012245230&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012245230&partnerID=8YFLogxK

U2 - 10.4208/jcm.1603-m2015-0317

DO - 10.4208/jcm.1603-m2015-0317

M3 - Article

VL - 34

SP - 511

EP - 531

JO - Journal of Computational Mathematics

JF - Journal of Computational Mathematics

SN - 0254-9409

IS - 5

ER -