Communications on Applied Mathematics and Computation >

2022 , Vol. 4 >Issue 1: 180 - 204

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

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

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

Received date: 2020-09-05

  Revised date: 2020-11-10

  Online published: 2022-03-01

Supported by

Research work of Jue Yan is supported by the National Science Foundation grant DMS-1620335 and Simons Foundation Grant 637716. Research work of Xinghui 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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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

Cite this article

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 . DOI: 10.1007/s42967-020-00107-0

References

1. Adjerid, S., Devine, K., Flaherty, J., Krivodonova, L.:A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191(11/12), 1097-1112 (2002)
2. Ainsworth, M.:Dispersive and dissipative behaviour of high order discontinuous Galerkin fnite element methods. J. Comput. Phys. 198(1), 106-130 (2004)
3. Ainsworth, M., Monk, P., Muniz, W.:Dispersive and dissipative properties of discontinuous Galerkin fnite element methods for the second-order wave equation. J. Sci. Comput. 27(1), 5-40 (2006)
4. Arnold, D.N.:An interior penalty fnite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742-760 (1982)
5. Baumann, C.E., Oden, J.T.:A discontinuous hp fnite element method for convection-difusion problems. Comput. Methods Appl. Mech. Eng. 175(3/4), 311-341 (1999)
6. Cao, W., Zhang, Z.:Superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. Math. Comput. 85, 63-84 (2014)
7. Cao, W., Liu, H., Zhang, Z.:Superconvergence of the direct discontinuous Galerkin method for convectiondifusion equations. Numer. Methods Partial Difer. Equ. 33(1), 290-317 (2017)
8. Chen, Z., Huang, H., Yan, J.:Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection difusion equations on unstructured triangular meshes. J. Comput. Phys. 308, 198-217 (2016)
9. Cheng, Y., Shu, C.-W.:Superconvergence and time evolution of discontinuous Galerkin fnite element solutions. J. Comput. Phys. 227(22), 9612-9627 (2008)
10. Cheng, Y., Shu, C.-W.:Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-difusion equations. Comput. Struct. 87, 630-641 (2009)
11. Chuenjarern, N., Yang, Y.:Fourier analysis of local discontinuous Galerkin methods for linear parabolic equations on overlapping meshes. J. Sci. Comput. 81, 671-688 (2019)
12. Cockburn, B., Shu, C.-W.:The local discontinuous Galerkin method for time-dependent convection-difusion systems. SIAM J. Numer. Anal. 35(6), 2440-2463 (1998). Electronic
13. Cockburn, B., Luskin, M., Shu, C.-W., Suli, E.:Enhanced accuracy by post-processing for fnite element methods for hyperbolic equations. Math. Comput. 72(242), 577-606 (2003)
14. Guo, W., Zhong, X., Qiu, J.-M.:Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods:eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235, 458-485 (2013)
15. He, Y., Li, F., Qiu, J.:Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput. 55(3), 552-574 (2013)
16. Hu, F., Hussaini, M., Rasetarinera, P.:An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151(2), 921-946 (1999)
17. Huang, H., Li, J., Yan, J.:High order symmetric direct discontinuous Galerkin method for elliptic interface problems with ftted mesh. J. Comput. Phys. 409, 109301-109323 (2020)
18. Ji, L., Xu, Y., Ryan, J.:Accuracy-enhancement of discontinuous Galerkin solutions for convection-difusion equations in multiple-dimensions. Math. Comput. 81(280), 1929-1950 (2012)
19. Liu, H., Yan, J.:The direct discontinuous Galerkin (DDG) methods for difusion problems. SIAM J. Numer. Anal. 47(1), 475-698 (2009)
20. Liu, H., Yan, J.:The direct discontinuous Galerkin (DDG) method for difusion with interface corrections. Commun. Comput. Phys. 8(3), 541-564 (2010)
21. Rivière, B., Wheeler, M.F., Girault, V.:A priori error estimates for fnite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902-931 (2001). Electronic
22. Sármány, D., Botchev, M., van der Vegt, J.:Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations. J. Sci. Comput. 33(1), 47-74 (2007)
23. Sherwin, S.:Dispersion analysis of the continuous and discontinuous Galerkin formulation. In:Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 425-432. Springer, Berlin, Heidelberg (2000)
24. Shu, C.-W., Osher, S.:Efcient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988)
25. Vidden, C., Yan, J.:A new direct discontinuous Galerkin method with symmetric structure for nonlinear difusion equations. J. Comput. Math. 31(6), 638-662 (2013)
26. Wheeler, M.F.:An elliptic collocation-fnite element method with interior penalties. SIAM J. Numer. Anal. 15, 152-161 (1978)
27. Yan, J.:A new nonsymmetric discontinuous Galerkin method for time dependent convection difusion equations. J. Sci. Comput. 54(2), 663-683 (2013)
28. Yang, Y., Shu, C.-W.:Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110-3133 (2012)
29. Yang, Y., Shu, C.-W.:Analysis of sharp superconvergence of local discontinuous Galerkin method for onedimensional linear parabolic equations. J. Comput. Math. 33, 323-340 (2015)
30. Zhang, M., Shu, C.-W.:An analysis of three diferent formulations of the discontinuous Galerkin method for difusion equations. Math. Models Methods Appl. Sci. 13(3), 395-413 (2003)
31. Zhang, M., Shu, C.-W.:Fourier analysis for discontinuous Galerkin and related methods. Sci. Bull. 54(11), 1809-1816 (2009)
32. Zhang, M., Yan, J.:Fourier type error analysis of the direct discontinuous Galerkin method and its variations for difusion equations. J. Sci. Comput. 52(3), 638-655 (2012)
33. Zhang, M., Yan, J.:Fourier type super convergence study on DDGIC and symmetric DDG methods. J. Sci. Comput. 73(2/3), 1276-1289 (2017)
34. Zhang, Y., Zhang, X., Shu, C.-W.:Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-difusion equations on triangular meshes. J. Comput. Phys. 234, 295-316 (2013)
35. Zhong, X., Shu, C.-W.:Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200(41/42/43/44), 2814-2827 (2011)
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