TY - JOUR
T1 - Asymptotically exact a posteriori error estimates for the local discontinuous galerkin method for nonlinear kdv equations in one space dimension
AU - Baccouch, Mahboub
N1 - Funding Information:
Funding: This study was funded by the University Committee on Research and Creative Activity (UCRCA Proposal 2018-01-F) at the University of Nebraska at Omaha.
Funding Information:
This study was funded by the University Committee on Re-search and Creative Activity (UCRCA Proposal 2018-01-F) at the University of Nebraska at Omaha.
Publisher Copyright:
© 2020 Institute for Scientific Computing and Information.
PY - 2020
Y1 - 2020
N2 - In this paper, we develop and analyze an implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the (p+1)-degree right Radau polynomial and the second part converges with order p +3 in the2L2-norm, when piecewise polynomials of degree at most p are used. These results allow us to construct a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge at a fixed time to the exact spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p +3 . Finally, we prove that 2 the global effectivity index converges to unity at O(h12 ) rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.
AB - In this paper, we develop and analyze an implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the (p+1)-degree right Radau polynomial and the second part converges with order p +3 in the2L2-norm, when piecewise polynomials of degree at most p are used. These results allow us to construct a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge at a fixed time to the exact spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p +3 . Finally, we prove that 2 the global effectivity index converges to unity at O(h12 ) rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.
KW - A posteriori error estimation
KW - Local discontinuous Galerkin method
KW - Nonlinear KdV equations
KW - Supercon-vergence
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M3 - Article
AN - SCOPUS:85117302536
SN - 1705-5105
VL - 17
SP - 767
EP - 793
JO - International Journal of Numerical Analysis and Modeling
JF - International Journal of Numerical Analysis and Modeling
IS - 6
ER -