Advertisement

Journal of Scientific Computing

, Volume 77, Issue 3, pp 1490–1518 | Cite as

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

  • Jie Du
  • Eric Chung
Article
  • 93 Downloads

Abstract

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.

Keywords

Convection–diffusion Staggered discontinuous Galerkin method Error indicator a-posteriori error estimate Adaptive refinement 

Notes

Acknowledgements

The work of Eric Chung is partially supported by Hong Kong RGC General Research Fund (Projects: 14317516, 14301314) and CUHK Direct Grant for Research 2016-17.

References

  1. 1.
    Ahmed, N., Matthies, G.: Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent linear convection–diffusion-reaction equations. J. Sci. Comput. 67, 998–1018 (2015)MathSciNetGoogle Scholar
  2. 2.
    Ahmed, N., Matthies, G., Tobiska, L., Xie, H.: Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 200, 1747–1756 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ayuso, B., Marini, L.D.: Discontinuous Galerkin methods for advection–diffusion-reaction problems. SIAM J. Numer. Anal. 47, 1391–1420 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Braack, M., Lube, G.: Finite elements with local projection stabilization for incompressible flow problems. J. Comput. Math. 27, 116–147 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43, 2012–2033 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burman, E., Ern, A.: Continuous interior penalty hp-finite element methods for advection and advection–diffusion equations. Math. Comput. 76, 1119–1140 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cangiani, A., Georgoulis, E.H., Metcalfe, S.: Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems. IMA J. Numer. Anal. 34(4), 1578–1597 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, H., Li, J., Qiu, W.: Robust a posteriori error estimates for HDG method for convection–diffusion equations. IMA J. Numer. Anal. 36, 437–462 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, H., Qiu, W., Shi, K.: A priori and computable a posteriori error estimates for an HDG method for the coercive Maxwell equations. Comput. Methods Appl. Mech. Eng. 333, 287–310 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cheung, S.W., Chung, E., Kim, H.H., Qian, Y.: Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 302, 251–266 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chung, E.T., Ciarlet Jr., P.: A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials. J. Comput. Appl. Math. 239, 189–207 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chung, E.T., Ciarlet Jr., P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. Part II: the Stokes flow. J. Sci. Comput. 66, 870–887 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chung, E.T., Du, J., Yuen, M.C.: An adaptive SDG method for the Stokes system. J. Sci. Comput. 70, 766–792 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131–2158 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chung, E.T., Kim, H.H., Widlund, O.: Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51, 47–67 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the convection–diffusion equation. J. Numer. Math. 20, 1–31 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chung, E.T., Leung, W.T.: A sub-grid structure enhanced discontinuous Galerkin method for multiscale diffusion and convection–diffusion problems. Commun. Comput. Phys. 14, 370–392 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chung, E., Yuen, M.C., Zhong, L.: A-posteriori error analysis for a staggered discontinuous Galerkin discretization of the time-harmonic Maxwell’s equations. Appl. Math. Comput. 237, 613–631 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Cockburn, B., Dong, B., Guzman, J., Restelli, M., Sacco, R.: A hybridizable discontinuous Galerkin method for steady-state convection–diffusion-reaction problems. SIAM J. Sci. Comput. 31, 3827–3846 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Codina, R.: Finite element approximation of the convection-diffusion equation: subgrid-scale spaces, local instabilities and anisotropic space-time discretizations. Lecture Notes in Computational Science and Engineering, vol. 81, pp. 85–97 (2011)Google Scholar
  26. 26.
    Dörfler, W.: A convergent adaptive algorithm for Poissons equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ern, A., Guermond, J.: Theory and Practice of Finite Elements. Applied mathematical sciences. Springer, New York (2004)CrossRefGoogle Scholar
  28. 28.
    Ern, A., Stephansen, A.F., Vohralik, M.: Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection–diffusion reaction problems. J. Comput. Appl. Math. 234, 114–130 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Fu, G., Qiu, W., Zhang, W.: An analysis of HDG methods for convection dominated diffusion problems. ESAIM Math. Model. Numer. Anal. 49, 225–256 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Houston, P., Perugia, I., Schotzau, D.: An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations. IMA J. Numer. Anal. 27, 122–150 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Matthies, G., Skrzypacz, P., Tobiska, L.: Stabilization of local projection type applied to convection–diffusion problems with mixed boundary conditions. Electron. Trans. Numer. Anal. 32, 90–105 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nguyen, N., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. J. Comput. Phys. 228, 3232–3254 (2009)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Qiu, W., Shi, K.: An HDG method for convection diffusion equation. J. Sci. Comput. 66, 346–357 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rostand, V., Le Roux, D.Y.: Raviart–Thomas and Brezzi–Douglas–Marini finite-element approximations of the shallow-water equations. Int. J. Numer. Methods Fluids 57, 951–976 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Süli, E., Schwab, C., Houston, P.: hp-DGFEM for partial differential equations with nonnegative characteristic form. In: Cockburn, B., Karniadakis, G. E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol. . Springer, Berlin, pp. 221–230 (2000)Google Scholar
  40. 40.
    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Vohralik, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of converction–diffusion-reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsThe Chinese University of Hong KongSha TinHong Kong SAR

Personalised recommendations