A posteriori local discontinuous Galerkin error estimates for the one-dimensional sine-Gordon equation

In this paper, we present a posteriori error analysis of the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equations with smooth solutions. We show that the dominant components of the local LDG errors on each element are proportional to right and left Radau polynomials of degree p+1. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are (Formula presented.) superconvergent at the roots of (Formula presented.) -degree right and left Radau polynomials, respectively. Numerical experiments indicate that our superconvergence results hold globally. We use the superconvergence results to construct simple, efficient, and asymptotically exact a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative, at any fixed time, converge to the true errors at (Formula presented.) rate, respectively. We also demonstrate that the global effectivity indices for the solution and its derivative in the (Formula presented.) -norm converge to unity. We present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimates.