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Hui Bi, Yanan Xu, Yang Sun. DIFFERENCE QUOTIENT ESTIMATES AND ACCURACY ENHANCEMENT OF DISCONTINUOUS GALERKIN METHODS FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1766-1796. doi: 10.11948/20220012
Citation: Hui Bi, Yanan Xu, Yang Sun. DIFFERENCE QUOTIENT ESTIMATES AND ACCURACY ENHANCEMENT OF DISCONTINUOUS GALERKIN METHODS FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1766-1796. doi: 10.11948/20220012
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DIFFERENCE QUOTIENT ESTIMATES AND ACCURACY ENHANCEMENT OF DISCONTINUOUS GALERKIN METHODS FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS

  • 1.

    Applied Mathematics Department, Harbin University of Science and Technology, Xuefu Road, 10080 Harbin, China

  • Author Bio: Email: xuyanan0114@163.com(Y. Xu); Email: sunsy@126.com(Y. Sun)
  • Corresponding author: Email: bihui@hrbust.edu.cn (H. Bi) 
  • Fund Project: The work was supported by National Natural Science Foundation of China (11801122), National Science Foundation of Heilongjiang (LH2020A015) and the Fundamental Research Fundation for Universities of Heilongjiang Province (LGYC2018JC001)
    Abstract

  • In this paper, we apply the post-processing technique to the improvement of the superconvergence of the discontinuous Galerkin method for the nonlinear convection-diffusion equations. We firstly analyze the error estimate and convergence accuracy under $ L^{2} $-norm, and then demonstrate that the $ \alpha $-order difference quotient of DG error is of order $ k+3/2-\alpha/2 $ when the upwind fluxes are used. By the duality argument, we construct an appropriate dual equation, and futher obtain superconvergence results of order in the negative-order norm, namely $ 2k+3/2-\alpha/2 $ order superconvergence accuracy. Finally, we choose an appropriate kernel function and apply the SIAC filter to the nonlinear convection-diffusion equation to obtain at least $ 3k/2+1 $ order superconvergence for post-processed solutions. All theoretical results are proved by numerical experiments.

    MSC: 65M12, 65M15, 65M60
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