### Abstract

In this paper, new a posteriori error estimates for the local discontinuous Galerkin (LDG) formulation applied to transient convection-diffusion problems in one space dimension are presented and analyzed. These error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree right Radau polynomial while the leading error term for the solution's derivative is proportional to a (p+1)-degree left Radau polynomial, when polynomials of degree at most p are used. These results are used to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the ^{L2}-norm under mesh refinement. More precisely, we prove that our LDG error estimates converge to the true spatial errors at O(hp+^{5/4}) rate. Finally, we prove that the global effectivity indices in the ^{L2}-norm converge to unity at O(h1^{/2}) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

Original language | English (US) |
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Pages (from-to) | 455-483 |

Number of pages | 29 |

Journal | Applied Mathematics and Computation |

Volume | 226 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics