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Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems

Published online by Cambridge University Press:  28 May 2015

Tie Zhang  and
Jingna Liu
Tie Zhang*
Affiliation:
Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China
Jingna Liu
Affiliation:
Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China
*
Corresponding author.Email address:[email protected]
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Abstract

We study the enhancement of accuracy, by means of the convolution post-processing technique, for discontinuous Galerkin(DG) approximations to hyperbolic problems. Previous investigations have focused on the superconvergence obtained by this technique for elliptic, time-dependent hyperbolic and convection-diffusion problems. In this paper, we demonstrate that it is possible to extend this post-processing technique to the hyperbolic problems written as the Friedrichs’ systems by using an upwind-like DG method. We prove that the L2-error of the DG solution is of order k+1/2, and further the post-processed DG solution is of order 2k+1 if Qk-polynomials are used. The key element of our analysis is to derive the (2k+1)-order negative norm error estimate. Numerical experiments are provided to illustrate the theoretical analysis.

Keywords

65N3065M60Discontinuous Galerkin methodhyperbolic problemaccuracy enhancementpost-processingnegative norm error estimate
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 2 , May 2014 , pp. 214 - 233
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Bramble, J.H. and Schatz, A.H., Higher order local accuracy by averaging in the finite element method, Mathematics of Computation, 31 (1977), pp. 94–111.Google Scholar
[2] Cockburn, B., Luskin, M., Shu, C.W. and Süli, E., Enhanced accuracy by podt-processing for finite element methods for hyperbolic equations, Mathematics of Computation, 72 (2003), pp. 577–606.Google Scholar
[3] Cockburn, B., Karniadakis, G.E. and Shu, C.W., Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes Comput. Sci. Eng., Vol. 11, Springer-Verlag, Berlin, 2000.Google Scholar
[4] Cockburn, B., Dong, B. and Guzméan, J., Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal., 46 (2008), pp. 1250–1265.Google Scholar
[5] Ern, A. and J.Guermond, L., Discontinuous Galerkin methods for Friedrichs’ systems, I. general theory, SIAM J. Numer. Anal., 44 (2006), pp. 753–778.Google Scholar
[6] Falk, R.S. and Richter, G.R., Explicit finite element methods for symmetric hyperbolic equations, SIAM J. Numer. Anal., 36 (1999), pp. 935–952.Google Scholar
[7] Friedrichs, K., Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), pp. 333–418.CrossRefGoogle Scholar
[8] Johnson, C., U. Nävert, and Pitkaranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 285–312.CrossRefGoogle Scholar
[9] Ji, L.Y., Xu, Y. and Ryan, J.K., Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions, Mathematics of Computation, 81 (2012), pp. 1929–1950.CrossRefGoogle Scholar
[10] Krizek, M., Neittaanmaki, P. and Stenberg, R., Finite Element Methods: Superconvergence, Postprocessing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series, Vol. 196, Dekker, Inc., New York, 1997.Google Scholar
[11] Lin, Q. and Zhu, Q.D., The Preprocessing and Postprocessing for the Finite Element Methods, Shanghai Sci. & Tech. Publishing, Shanghai, China, 1994.Google Scholar
[12] Lesaint P, P. and Raviart, P.A., On a finite element method for solving the neutron transport equation, in: Mathematical Aspects of Finite Elements in Partial Differential Equations, de Boor, C., ed. Academic Press, New York, 1974, pp. 89–145.Google Scholar
[13] Monk, P. and Richter, G.R., A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Compu., 22 (2005), pp. 443–477.Google Scholar
[14] Mirzaee, H., Ji, L., Ryan, J.K. and Kirby, R.M., Smoothness-increasing accuracy-conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes, SIAM Journal on Numerical Analysis, 49 (2011), pp. 1899–1920.Google Scholar
[15] Peterson, T.E., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), pp. 133–140.Google Scholar
[16] Richter, G., An optimal-order error estimate for discontinuous Galerkin method, Math. Comp., 50 (1988), pp. 75–88.Google Scholar
[17] Rauch, J., L2 is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure Appl. Math. 25 (1972), pp. 265–285.Google Scholar
[18] Ryan, J.K., Shu, C.W. and Atkins, H., Extension of a post-processing technique for the discontinuous Galerkin method for hyperbolic equations with application to an aeroacoustic problem, SIAM J. Sci. Comput., 26 (2005), pp. 821–843.Google Scholar
[19] Vila, J.P. and Villedieu, P., Convergence of an explicit finite volume scheme for first order symmetric systems, Numer. Math., 94 (2003), pp. 573–602.CrossRefGoogle Scholar
[20] Zienkiewicz, O.C. and Zhu, J.Z., The superconvergence patch recovery and a posterior error estimates, Part 1: the recovery technique, Int. J. Numer. Methods Engrg. 33 (1992), pp. 1331–1364.Google Scholar
[21] Zhang, Z., Ultraconvergence of the patch recovery technique II, Math. Comput. 69 (2000), pp. 141–158.Google Scholar
[22] Zhang, T., Discontinuous finite element method for first order hyperbolic system of equations, J. of Northeastern Univ., 22 (1987), pp. 250–257.Google Scholar