Science China Mathematics

, Volume 57, Issue 11, pp 2301–2320 | Cite as

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

Articles
First Online:
Received:
Accepted:

Abstract

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L ∞(0, T;L 2) for concentration c, in L 2(0, T; L 2) for c x and L ∞(0, T;L 2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally, we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.

Keywords

mixed finite element method local discontinuous Galerkin method error estimate miscible displacement problem 

MSC(2010)

65M15 65M60 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arbogast T, Wheeler M F, Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J Numer Anal, 1997, 34: 828–852MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold D N, Brezzi F, Cockburn B, et al. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal, 2002, 39: 1749–1779MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ayuso B, Carrillo J A, Shu C W. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinet Relat Models, 2011, 4: 955–989MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bassi F, Rebay S. A high-order accurate discontinuous finite element method for the numerical solution to the compressible Navier-Stokes equations. J Comput Phys, 1997, 131: 267–279MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bear J. Dynamics of Fluids in Porous Media. New York: Dover Publications Inc, 1972MATHGoogle Scholar
  6. 6.
    Castillo P, Cockburn B, Perugia I, et al. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J Numer Anal, 2000, 38: 676–706MathSciNetCrossRefGoogle Scholar
  7. 7.
    Castillo P, Cockburn B, Schötzau D, et al. Optimal a priori error estimates for the hp-version of the LDG method for convection diffusion problems. Math Comp, 2002, 71: 455–478MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ciarlet P. The Finite Element Method for Elliptic Problem. Amsterdam: North Holland, 1975Google Scholar
  9. 9.
    Cockburn B, Hou S, Shu C W. The Runge C Kutta local projection discontinuous Galerkin finite element method for conservation laws, IV: The multidimensional case. Math Comp, 1990, 54: 545–581MathSciNetMATHGoogle Scholar
  10. 10.
    Cockburn B, Lin S Y, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, III: One-dimensional systems. J Comput Phys, 1989, 84: 90–113MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cockburn B, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, II: General framework. Math Comp, 1989, 52: 411–435MathSciNetMATHGoogle Scholar
  12. 12.
    Cockburn B, Shu C W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal, 1998, 35: 2440–2463MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cockburn B, Shu C W. The Runge-Kutta discontinuous Galerkin method for conservation laws, V: Multidimensional systems. J Comput Phys, 1998, 141: 199–224MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Douglas J J, Ewing R E, Wheeler M F. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Numer Anal, 1983, 17: 17–33MathSciNetMATHGoogle Scholar
  15. 15.
    Douglas J J, Ewing R E, Wheeler M F. A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Numer Anal, 1983, 17: 249–265MathSciNetMATHGoogle Scholar
  16. 16.
    Douglas J J, Roberts J E. Numerical methods for a model for compressible miscible displacement in porous media. Math Comp, 1983, 41: 441–459MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Douglas J J, Roberts J E. Global Estimates for Mixed Methods for Second Order Elliptic Equations. Math Comp, 1985, 44: 39–52MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dullien F A L. Porous Media Fluid Transport and Pore Structure. New York: Academic Press, 1979Google Scholar
  19. 19.
    Liu Y X, Shu C W. Error analysis of the semi-discrete local discontinuous Galerkin method for semiconductor device simulation models. Sci China Math, 2010, 53: 3255–3278MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Reed W H, Hill T R. Triangular Mesh Method for the Neutron Transport Equation. Technical report LA-UR-73-479. Los Alamos, NM: Los Alamos Scientific Laboratory, 1973Google Scholar
  21. 21.
    Shu C W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys, 1988, 77: 439–471MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sun S Y, Rivière B, Wheeler M F. A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media. In: Recent Progress in Computational and Applied PDEs, Kluwer Academic Publishers. Dordrecht-New York: Plenum Press, 2002, 323–351CrossRefGoogle Scholar
  23. 23.
    Sun S Y, Wheeler M F. Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl Numer Math, 2005, 52: 273–298MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Yan J, Shu C W. A local discontinuous Galerkin method for KdV type equations. SIAM J Numer Anal, 2002, 40: 769–791MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Xu Y, Shu C W. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J Comput Phys, 2005, 205: 72–97MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Xu Y, Shu C W. Local discontinuous Galerkin methods for the Kuramoto C Sivashinsky equations and the Ito-type coupled KdV equations. Comput Method Appl Mech Eng, 2006, 195: 3430–3447MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Xu Y, Shu C W. Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J Numer Anal, 2012, 50: 79–104MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zhang Q, Gao F Z. A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J Sci Comput, 2012, 51: 107–134MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoChina
  2. 2.Michigan Technological UniversityHoughtonUSA

Personalised recommendations