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Journal of Scientific Computing

, Volume 70, Issue 2, pp 744–765 | Cite as

Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations

  • Hongying Huang
  • Zheng Chen
  • Jin Li
  • Jue YanEmail author
Article

Abstract

In this paper, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under \(L^2\) norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal \((k+1)\)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal \((k+1)\)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

Keywords

Discontinuous Galerkin method Second order elliptic problem 

Notes

Acknowledgments

Huang’s work is supported by Natural Science Foundation of Zhejiang Province Grant Nos. LY14A010002 and LY12A01009, and is subsidized by the National Natural Science Foundation of China under Grant Nos. 11101368, 11471195, 11001168 and 11202187. Yan’s research is supported by the US National Science Foundation (NSF) under grant DMS-1620335.

References

  1. 1.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baumann, C.E., Oden, J.T.: A discontinuous \(hp\) finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Owens, L., Sung, L.-Y.: A weakly over-penalized symmetric interior penalty method. Electron. Trans. Numer. Anal. 30, 107–127 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)CrossRefGoogle Scholar
  8. 8.
    Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E.: Advanced numerical approximation of nonlinear hyperbolic equations, volume 1697 of Lecture Notes in Mathematics. In: Quarteroni, A. (ed.) Papers from the C.I.M.E. Summer School Held in Cetraro, 23–28 June 1997. Springer-Verlag, Berlin (1998). Fondazione C.I.M.E. [C.I.M.E. Foundation]Google Scholar
  11. 11.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012)Google Scholar
  14. 14.
    Ewing, R., Iliev, O., Lazarov, R.: A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients. SIAM J. Sci. Comput. 23(4), 1335–1351 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guzmán, J., Rivière, B.: Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations. J. Sci. Comput. 40(1–3), 273–280 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods, Volume 54 of Texts in Applied Mathematics. Springer, New York (2008). (Algorithms, analysis, and applications)CrossRefGoogle Scholar
  17. 17.
    Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 475–698 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)MathSciNetGoogle Scholar
  19. 19.
    Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous \(hp\) finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math., Vol. 606, pp. 292–315. Springer, Berlin (1977)Google Scholar
  21. 21.
    Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations Volume 35 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008). (Theory and implementation)CrossRefGoogle Scholar
  22. 22.
    Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902–931 (2001). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shu, C-w: Discontinuous Galerkin method for time-dependent problems: survey and recent developments. In: Recent developments in discontinuous Galerkin finite element methods for partial differential equations, volume 157 of IMA Vol. Math. Appl., pp. 25–62. Springer, Cham (2014)Google Scholar
  24. 24.
    Vidden, C., Yan, J.: A new direct discontinuous Galerkin method with symmetric structure for nonlinear diffusion equations. J. Comput. Math. 31(6), 638–662 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yan, J.: A new nonsymmetric discontinuous Galerkin method for time dependent convection diffusion equations. J. Sci. Comput. 54(2–3), 663–683 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematics, Physics and Information ScienceZhejiang Ocean UniversityZhoushanChina
  2. 2.Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang ProvinceZhoushanChina
  3. 3.Department of MathematicsIowa State UniversityAmesUSA
  4. 4.School of ScienceShandong Jianzhu UniversityJinanChina

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