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
N1 - Publisher Copyright:
Copyright 2016 by AMSS, Chinese Academy of Science.
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
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U2 - 10.4208/jcm.1603-m2015-0317
DO - 10.4208/jcm.1603-m2015-0317
M3 - Article
AN - SCOPUS:85012245230
SN - 0254-9409
VL - 34
SP - 511
EP - 531
JO - Journal of Computational Mathematics
JF - Journal of Computational Mathematics
IS - 5
ER -