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 $k\geq 2$, the DDG solution is superconvergent at nodes and Lobatto points, with an order of ${\cal O}(h^{2k})$ and ${\cal O}(h^{k+2})$, respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of ${\cal O}(h^{k+1})$. Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

翻译：本文涉及直接不连续的 Galerkin (DDG) 方法对二维非线性对流扩散方程式的超趋同性特性。 此外,通过使用校正函数的理念,我们证明,对于任何片段的 Exor 产品多元度( $k\geq 2), DDG 解决方案在节点和Lobatto 点具有超级趋同性, 其顺序分别为$\cal O}( h ⁇ 2k}) $ 和 ${cal O} (h ⁇ k+2}) 。 此外, 还对衍生物近似值的超趋同性特性进行了研究, 并在高斯点确定了超级趋同点, 其顺序为$\cal O} (h ⁇ k+1} 。 数字实验证实了所有理论结论的清晰度 。