A posteriori error analysis of the local discontinuous Galerkin method for the sine–Gordon equation in one space dimension

Abstract

In this paper, we present asymptotically exact a posteriori error estimators of the local discontinuous Galerkin (LDG) method for the nonlinear sine–Gordon equation in one space dimension. We apply our recent optimal L2 error estimates and superconvergence results (Baccouch, 2018) to show that the LDG errors on each element can be split into two parts. The first part is proportional to the (p+1)-degree Radau polynomial and the second part converges with order p+3∕2 in the L2-norm, when piecewise polynomials of degree at most p are used. This new result is used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. We further prove that, for smooth solutions, these a posteriori error estimates for the solution and its derivative, at a fixed time, converge to the exact spatial errors in the L2-norm under mesh refinement. The theoretical order of convergence is proved to be p+3∕2 while our computational results show higher O(hp+2) convergence rate. Finally, we prove that the global effectivity index converges to unity at O(h1∕2) rate. Several numerical results are presented to validate the theoretical results.