Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation on Cartesian grids

In this paper, we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We perform a local error analysis and show that the actual error can be split into an O(hp+1) leading component and a higher-order component, when tensor product polynomials of degree at most p are used. We further prove that the leading term of the LDG error is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. Thus, the LDG solution is O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Computational results indicate that our superconvergence results hold globally. We use these 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. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.