In this paper,we apply the Fourier analysis technique to investigate superconvergence prop-erties 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 inte-rior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quanti-tative errors at the Lobattto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P2 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China;Department of Mathematics,Iowa State University,Ames,IA 50011,USA

[1]Yuqing Miao;Jue Yan;Xinghui Zhong-.Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations)[J].应用数学与计算数学学报,2022(01):180-204

Superconvergence,disontinuous,DDG,SIAM,DDGIC,Comput,Vidden,IPDG,Lobattto,mations

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