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arXiv · 2111.03986

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

Xinyue Zhang·Waixiang Cao
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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.

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Xinyue Zhang, Waixiang Cao. 2021-11-07. Superconvergence of the Direct Discontinuous Galerkin Method for Two-Dimensional Nonlinear Convection-Diffusion Equations. https://arxiv.org/abs/2111.03986

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