TY - JOUR

T1 - Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems

AU - Baccouch, Mahboub

N1 - Funding Information:
The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Local discontinuous Galerkin method

KW - Projections

KW - Radau points

KW - Superconvergence

KW - Transient convection-diffusion problems

KW - a posteriori error estimation

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U2 - 10.1016/j.amc.2013.10.026

DO - 10.1016/j.amc.2013.10.026

M3 - Article

AN - SCOPUS:84888242405

VL - 226

SP - 455

EP - 483

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -