Gradient recovery based a posteriori error estimator for the adaptive direct discontinuous Galerkin method

In this paper, we propose a gradient recovery method for the direct discontinuous Galerkin (DDG) method. A quadratic polynomial is obtain by using the local discrete least-squares fitting to the gradient of numerical solution at certain sampling points. The recovered gradient is defined on a piecewise continuous space, and it may be discontinuous on the whole domain. Based on the recovered gradient, we introduce a posteriori error estimator which takes the L2 norm of the difference between the direct and post-processed approximations. Some benchmark test problems with typical difficulties are carried out to illustrate the superconvergence of the recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator. Most of the test problems are from the US National Institute for Standards and Technology (NIST).