A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD FOR SLIGHTLY COMPRESSIBLE FLOWS IN POROUS MEDIA

- Journal title : Journal of the Korean Mathematical Society
- Volume 44, Issue 5, 2007, pp.1103-1119
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2007.44.5.1103

Title & Authors

A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD FOR SLIGHTLY COMPRESSIBLE FLOWS IN POROUS MEDIA

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

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

Abstract

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.

Keywords

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

Language

English

Cited by

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References

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.

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

3.

J. Bear, Dynamics of Fluids in Porous Media, Dover Publication, Inc. New York, 1988

4.

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

5.

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

6.

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

7.

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991

8.

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

9.

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

10.

L. C. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64 (1995), no. 211, 989-1015

11.

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

12.

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

13.

R. Duran, Superconuerqence for rectangular mixed finite elements, Numer. Math. 58 (1990), no. 3, 287-298

14.

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

15.

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

16.

P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985

17.

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

18.

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)

19.

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

20.

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

21.

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

23.

E.-J. Park, Mixed finite element methods for nonlinear second-order elliptic problems, SIAM J. Numer. Anal. 32 (1995), no. 3, 865-885

24.

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

25.

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

26.

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

27.

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

28.

L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190,483-493

29.

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

30.

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

31.

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

32.

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