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

TL;DR

This paper proves that the direct discontinuous Galerkin method exhibits superconvergence at specific points for 2D nonlinear convection-diffusion equations, with numerical results confirming the theoretical orders of accuracy.

Contribution

It establishes superconvergence properties of the DDG method for 2D nonlinear convection-diffusion equations using correction functions, with new superconvergence orders at nodes, Lobatto, and Gauss points.

Findings

Superconvergence at nodes with order ${\

}h^{2k}$.

Superconvergence at Lobatto points with order ${\

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 $k\geq 2$, 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.