Optimal superconvergence and asymptotically exact a posteriori error estimator for the local discontinuous Galerkin method for linear elliptic problems on Cartesian grids

The purpose of this paper is twofold: to study the superconvergence properties and present an efficient reliable a posteriori error estimator for local discontinuous Galerkin (LDG) method linear second-order elliptic problems on Cartesian grids. We prove that LDG solution superconvergent towards particular projection exact solution. order convergence proved be p + 2 , when tensor product polynomials degree at most are used. Then, we actual can split into two parts. components significant part given in terms ( 1 ) -degree Radau polynomials. use these results construct residual-type estimates. proposed estimates converge true errors L -norm under mesh refinement. . Finally, AMR procedure makes our global provide several numerical examples illustrating effectiveness procedures.