Volume 5, Issue 3
Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method

MAHBOUB BACCOUCH

Int. J. Numer. Anal. Mod. B, 5 (2014), pp. 188-216

Published online: 2014-05

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  • Abstract
In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k+1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^{k+2}) superconvergent at the roots of (k+1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for P^k polynomials with k ≥ 1, and for various types of boundary conditions.
  • AMS Subject Headings

65M60 65N30 74K10

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COPYRIGHT: © Global Science Press

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@Article{IJNAMB-5-188, author = {MAHBOUB BACCOUCH}, title = { Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2014}, volume = {5}, number = {3}, pages = {188--216}, abstract = {In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k+1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^{k+2}) superconvergent at the roots of (k+1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for P^k polynomials with k ≥ 1, and for various types of boundary conditions.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/230.html} }
TY - JOUR T1 - Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method AU - MAHBOUB BACCOUCH JO - International Journal of Numerical Analysis Modeling Series B VL - 3 SP - 188 EP - 216 PY - 2014 DA - 2014/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnamb/230.html KW - Local discontinuous Galerkin method KW - fourth-order initial-boundary value problems KW - Euler-Bernoulli beam equation KW - superconvergence KW - a posteriori error estimates AB - In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k+1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^{k+2}) superconvergent at the roots of (k+1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for P^k polynomials with k ≥ 1, and for various types of boundary conditions.
MAHBOUB BACCOUCH. (1970). Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method. International Journal of Numerical Analysis Modeling Series B. 5 (3). 188-216. doi:
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