Skip to main content
Log in

Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript


We present an a priori stability and convergence analysis of a new mixed discontinuous Galerkin scheme applied to the instationary Darcy problem. The analysis accounts for a spatially and temporally varying permeability tensor in all estimates. The proposed method is stabilized using penalty terms in the primary and the flux unknowns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Aizinger, V.: A discontinuous Galerkin method for two-and three-dimensional shallow-water equations. Ph.D. thesis, The University of Texas at Austin. (2004)

  2. Aizinger, V., Dawson, C.: The local discontinuous Galerkin method for three-dimensional shallow water flow. Comput. Methods Appl. Mech. Eng. 196(4), 734–746 (2007). doi:10.1016/j.cma.2006.04.010.

    Article  Google Scholar 

  3. Aizinger, V., Dawson, C., Cockburn, B., Castillo, P.: The local discontinuous Galerkin method for contaminant transport. Adv. Water Resour. 24(1), 73–87 (2000). doi:10.1016/S0309-1708(00)00022-1

    Article  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Num. Anal. 39, 1749–1779 (2002)

    Article  Google Scholar 

  5. Barrios, T.P., Bustinza, R.: An a posteriori error analysis of an augmented discontinuous Galerkin formulation for Darcy flow. Numer. Math. 120(2), 231–269 (2012). doi:10.1007/s00211-011-0410-3

    Article  Google Scholar 

  6. Brezzi, F., Hughes, T., Marini, L., Masud, A.: Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput. 22-23, 119–145 (2005). doi:10.1007/s10915-004-4150-8

    Article  Google Scholar 

  7. Carrero, J., Cockburn, B., Schötzau, D.: Hybridized globally divergence-free LDG methods. Part I: The Stokes problem. Math. Comput. 75(254), 533–563 (2006). doi:10.1090/S0025-5718-05-01804-1 10.1090/S0025-5718-05-01804-1

    Article  Google Scholar 

  8. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2001). doi:10.1137/S0036142900371003

    Article  Google Scholar 

  9. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455–478 (2002). doi:10.1090/S0025-5718-01-01317-5

    Article  Google Scholar 

  10. Cheng, J., Shu, C.W.: High order schemes for CFD: a review. Jisuan Wuli/Chinese J. Comput. Phys. 26 (5), 633–655 (2009)

    Google Scholar 

  11. Ciarlet, P.G., Lions, J.L.: Handbook of numerical analysis. Elsevier (1990)

  12. Cockburn, B., Dawson, C.: Some Extensions of the Local Discontinuous Galerkin Method for Convection-Diffusion Equations in Multidimensions The Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, pp 225–238. Elsevier (2000)

  13. 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)

    Article  Google Scholar 

  14. Dawson, C.: The p k+1s k local discontinuous Galerkin method for elliptic equations. SIAM J. Numer. Anal. 40(6), 2151–2170 (2002). doi:10.1137/S0036142901397599

    Article  Google Scholar 

  15. Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods. Mathématiques et applications. Springer, Heidelberg (2012)

    Google Scholar 

  16. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2004)

    Book  Google Scholar 

  17. Hughes, T.J.R., Masud, A., Wan, J.: A stabilized mixed discontinuous Galerkin method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 195(25–28), 3347–3381 (2006). doi:10.1016/j.cma.2005.06.018

    Article  Google Scholar 

  18. Huynh, H.: A Flux Reconstruction Approach to High-Order Schemes including Discontinuous Galerkin Methods. In: Collection of Technical Papers - 18Th AIAA Computational Fluid Dynamics Conference, vol. 1, pp. 698–739. doi:10.2514/6.2007-4079 (2007)

  19. Knabner, P., Angermann, L.: Numerical methods for elliptic and parabolic partial differential equations. Springer (2003)

  20. Masud, A., Hughes, T.J.R.: A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 191(39-40), 4341–4370 (2002). doi:10.1016/S0045-7825(02)00371-7

    Article  Google Scholar 

  21. Nguyen, N., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods. Lecture Notes in Computational Science and Engineering 76 LNCSE, 63–84. doi:10.1007/978-3-642-15337-2_4 (2011)

  22. Peraire, J., Persson, P.O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30(4), 1806–1824 (2007). doi:10.1137/070685518

    Article  Google Scholar 

  23. Perugia, I., Schötzau, D.: An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17(1), 561–571 (2002). doi:10.1023/A:1015118613130

    Article  Google Scholar 

  24. Reed, H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory NM (1973)

  25. Shu, C.W.: A brief survey on discontinuous Galerkin methods in computational fluid dynamics. Adv. Mech. 43(6), 541–554 (2013). doi:10.6052/1000-0992-13-059

    Google Scholar 

  26. Shu, C.W.: High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments. J. Comput. Phys. 316, 598–613 (2016). doi:10.1016/

    Article  Google Scholar 

  27. Sun, S., Wheeler, M.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43(1), 195–219 (2005). doi:10.1137/S003614290241708X

    Article  Google Scholar 

  28. Sun, S., Wheeler, M.: Analysis of discontinuous Galerkin methods for multicomponent reactive transport problems. Comput. Math. Appl. 52(5), 637–650 (2006). doi:10.1016/j.camwa.2006.10.004

    Article  Google Scholar 

  29. Thomeé, V.: Galerkin Finite Element Methods for Parabolic Problems (2nd ed.) Springer (2006)

  30. Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput.Phys. 7(1), 1–46 (2010). doi:10.4208/cicp.2009.09.023 10.4208/cicp.2009.09.023

    Google Scholar 

  31. Zhang, M., Shu, C.W.: An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13(03), 395–413 (2003). doi:10.1142/S0218202503002568

    Article  Google Scholar 

  32. Zhang, X., Shu, C.W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: Survey and new developments. Proc. the Royal Soc. A: Math. Phys. Eng. Sci. 467(2134), 2752–2776 (2011). doi:10.1098/rspa.2011.0153

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Vadym Aizinger.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aizinger, V., Rupp, A., Schütz, J. et al. Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow. Comput Geosci 22, 179–194 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: