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Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

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Abstract

In this paper, we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009), the DDG method with the interface correction (DDGIC) (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010), the symmetric DDG method (Vidden and Yan in Comput Math 31(6):638–662, 2013), and the nonsymmetric DDG method (Yan in J Sci Comput 54(2):663–683, 2013). We also include the study of the interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both \(P^2\) and \(P^3\) polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that the DDGIC and symmetric DDG methods are superior, in the sense of obtaining \((k+2)\)th superconvergence orders for both \(P^2\) and \(P^3\) approximations. Superconvergence order of \((k+2)\) is also observed for the IPDG method with \(P^3\) polynomial approximations. The errors are sensitive to the choice of the numerical flux coefficient for even degree \(P^2\) approximations, but are not for odd degree \(P^3\) approximations. Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

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Acknowledgements

Research work of Jue Yan is supported by the National Science Foundation grant DMS-1620335 and Simons Foundation Grant 637716. Research work of **nghui Zhong is supported by the National Natural Science Foundation of China (NSFC) (Grant no. 11871428). The authors appreciate Dr. Waixiang Cao for many helpful discussions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Yuqing Miao & **nghui Zhong

  2. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Jue Yan

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  1. Yuqing Miao
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  2. Jue Yan
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Correspondence to **nghui Zhong.

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Miao, Y., Yan, J. & Zhong, X. Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations. Commun. Appl. Math. Comput. 4, 180–204 (2022). https://doi.org/10.1007/s42967-020-00107-0

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  • DOI: https://doi.org/10.1007/s42967-020-00107-0

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