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 to present an efficient and reliable a posteriori error estimator for the local discontinuous Galerkin (LDG) method for linear second-order elliptic problems on Cartesian grids. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved to be p+2, when tensor product polynomials of degree at most p are used. Then, we prove that the actual error can be split into two parts. The components of the significant part can be given in terms of (p+1)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We prove that the proposed residual-type a posteriori error estimates converge to the true errors in the L²-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. We provide several numerical examples illustrating the effectiveness of our procedures.