### 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 L^{2}-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 L^{2}-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 L^{2}-norm at O(h^{p}
^{+}
^{3}
^{/}
^{2}) rate. Finally, we prove that the global effectivity index in the L^{2}-norm converge to unity at O(h^{1 / 2}) rate. Our proofs are valid for arbitrary regular meshes using P^{p} polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.

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

Pages (from-to) | 1868-1904 |

Number of pages | 37 |

Journal | Journal of Scientific Computing |

Volume | 76 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2018 |

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### 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

**Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems

AU - Baccouch, Mahboub

PY - 2018/9/1

Y1 - 2018/9/1

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.

KW - Gauss–Radau projection

KW - Local discontinuous Galerkin method

KW - Nonlinear convection–diffusion problems

KW - Superconvergence

KW - a posteriori error estimation

UR - http://www.scopus.com/inward/record.url?scp=85050795067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050795067&partnerID=8YFLogxK

U2 - 10.1007/s10915-018-0687-9

DO - 10.1007/s10915-018-0687-9

M3 - Article

AN - SCOPUS:85050795067

VL - 76

SP - 1868

EP - 1904

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 3

ER -