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

doi:10.1007/s42967-020-00107-0

Abstract

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 P² and P³ 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 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 fux coefcient for even degree P² approximations, but are not for odd degree P³ approximations. Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

Key words: Direct discontinuous Galerkin methods, Superconvergence, Fourier analysis, Difusion equation

CLC Number:

65M60

Yuqing Miao, Jue Yan, Xinghui Zhong. Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Difusion Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 180-204.

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References

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