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
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 767
EP - 793
PY - 2020
DA - 2020/10
SN - 17
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/18350.html
KW - Local discontinuous Galerkin method, nonlinear KdV equations, superconvergence, $a$ $posteriori$ error estimation.
AB - <p style="text-align: justify;">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 $p$&nbsp;$+$ $\frac{3}{2}$ in the $L^2$-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 $L^2$-norm under
mesh refinement. The order of convergence is proved to be $p$ $+$&nbsp;$\frac{3}{2}$. Finally, we prove that
the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples
are provided to illustrate the global superconvergence results and the convergence of the
proposed error estimator.</p>