Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection-diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection-diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k >= 2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h(2k)) and O(h(k+2)), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(h(k+1)). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.