Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems

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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 languageEnglish (US)
Pages (from-to)455-483
Number of pages29
JournalApplied Mathematics and Computation
Volume226
DOIs
Publication statusPublished - 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

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