Superconvergence of the Direct Discontinuous Galerkin Method for Two-Dimensional Nonlinear Convection-Diffusion Equations

Abstract

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection-diffusion equations. By using the idea of correction function, we prove that, for any piecewise tensor-product polynomials of degree $k\ge 2$, the DDG solution is superconvergent at nodes and Lobatto points, with an order of $\mathcal{O}(h^{2k})$ and $\mathcal{O}(h^{k+2})$, respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of $\mathcal{O}(h^{k+1})$. Numerical experiments are presented to confirm the sharpness of all the theoretical findings.