Asymptotically exact a posteriori error estimates for the local discontinuous galerkin method for nonlinear kdv equations in one space dimension

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 the}2^L2-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.