• Kim, Mi-Young ;
  • Park, Eun-Jae ;
  • Thomas, Sunil G. ;
  • Wheeler, Mary F.
  • Published : 2007.09.30


We consider multiscale mortar mixed finite element discretizations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. Optimal fine scale convergence is obtained by an appropriate choice of mortar grid and polynomial degree of approximation. Parallel numerical simulations on some multiscale benchmark problems are given to show the efficiency and effectiveness of the method.


multiscale;mixed finite element;mortar finite element;error estimates;multiblock;non-matching grids


  1. T. Arbogast, L. C. Cowsar, M. F. Wheeler, and I. Yotov, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal. 37 (2000), no. 4, 1295-1315
  2. J. Bear, Dynamics of Fluids in Porous Media, Dover Publication, Inc. New York, 1988
  3. C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach. to domain decomposition: the mortar element method, Nonlinear partial differential equations and their applications. College de France Seminar, Vol. XI (Paris, 1989-1991), 13-51, Pitman Res. Notes Math. Ser., 299, Longman Sci. Tech., Harlow, 1994
  4. F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Model. Math. Anal. Numer. 21 (1987), no. 4, 581-604
  5. F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217-235
  6. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991
  7. Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp. 72 (2003), no. 242, 541-576
  8. M. A. Christie and M. J. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Reservoir Eval. Eng. 4 (2001), no. 4, 308-317
  9. L. C. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64 (1995), no. 211, 989-1015
  10. J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, The approxuruituni of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17 (1983), no. 1, 17-33
  11. R. Duran, Superconuerqence for rectangular mixed finite elements, Numer. Math. 58 (1990), no. 3, 287-298
  12. R. E. Ewing, R. D. Lazarov, and J. Wang, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal. 28 (1991), no. 4, 1015-1029
  13. R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), 144-172, SIAM, Philadelphia, PA,1988
  14. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985
  15. M.-Y. Kim, F. A. Milner, and E.-J. Park, Some observations on mixed methods for fully nonlinear parabolic problems in divergence form, Appl. Math. Lett. 9 (1996), no. 1, 75-81
  16. V. Kippe, J. E. Aarnes, and K.-A. Lie, Multiscale finite-element methods for elliptic problems in Porous media flow, Multiscale Model. Simul., (2006, to appear)
  17. T. P. Mathew, Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems, PhD thesis, Courant Institute of Mathematical Sciences, New York University, 1989
  18. F. A. Milner and E.-J. Park, A mixed finite element method for a strongly nonlinear second-order elliptic problem, Math. Comp. 64 (1995), no. 211, 973-988
  19. J. C. Nedelec, Mixed finite elements in $R^3$, Numer. Math. 35 (1980), no. 3, 315-341
  20. E.-J. Park, Mixed finite element methods for nonlinear second-order elliptic problems, SIAM J. Numer. Anal. 32 (1995), no. 3, 865-885
  21. E.-J. Park, Mixed finite element methods for generalized Forchheimer flow in porous media, Numer. Methods Partial Differential Equations 21 (2005), no. 2, 213-228
  22. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977
  23. J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of numerical analysis, Vol. II, 523-639, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991
  24. T. Russell and M. F. Wheeler, Finite element and finite difference methods for continuous flows in porous media in The Mathematics of Reservoir Simulation, R. E. Ewing, ed., Frontiers in Applied Mathematics 1, Society for Industrial and Applied Mathematics, Philadelphia, 1984, pp. 35-106
  25. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190,483-493
  26. A. Weiser and M. F. Wheeler, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), no. 2, 351-375
  27. M. F. Wheeler, A priori $L_2$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723-759
  28. M. F. Wheeler and I. Yotov, A posteriori error estimates for the mortar mixed finite element method, SIAM J. Numer. Anal. 43 (2005), no. 3, 1021-1042
  29. I. Yotov, Mixed finite element methods for flow in porous media, PhD thesis, Rice University, Houston, Texas, 1996, TR96-09, Dept. Compo Appl, Math., Rice University and TICAM report 96-23, University of Texas at Austin
  30. T. Arbogast, G. Pencheva, M. F. Wheeler, and I. Yotov, A multiscale mortar mixed finite element method, Technical Report TR-MATH 06-15, Department of Mathematics, University of Pittsburgh, 2006, Submitted to Multiscale Model. Simul
  31. J. Douglas, Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52
  32. M. Nakata, A. Weiser, and M. F. Wheeler, Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, The mathematics of finite elements and applications, V (Uxbridge, 1984), 367-389, Academic Press, London, 1985

Cited by

  1. Model coupling for multiphase flow in porous media vol.51, 2013,
  2. Coupling discontinuous Galerkin discretizations using mortar finite elements for advection–diffusion–reaction problems vol.67, pp.1, 2014,
  3. High-order discontinuous Galerkin methods with Lagrange multiplier for hyperbolic systems of conservation laws vol.73, pp.9, 2017,
  5. Stochastic collocation and mixed finite elements for flow in porous media vol.197, pp.43-44, 2008,
  6. A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions vol.70, pp.4, 2015,
  7. Mortar formulation for a class of staggered discontinuous Galerkin methods vol.71, pp.8, 2016,
  9. Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem 2017,