Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method

A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition(ϵ,σ). By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented.