Efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 3, 2008, Pages 314-336

  Younes, Anis ^a   Fontaine, Vincent ^b  

^a UNIVERSITÉ DE STRASBOURG   (France)

^b UNIVERSITÉ DE LA RÉUNION   (France)

Author keywords

Continuity point; Heterogeneous media; High anisotropy; Mixed hybrid finite element; Multipoint flux approximation; Quadrilateral mesh

Indexed keywords

ANISOTROPY; APPROXIMATION THEORY; POLYNOMIAL APPROXIMATION;

CONTINUITY POINT; HETEROGENEOUS MEDIA; HIGH ANISOTROPY; MIXED HYBRID FINITE ELEMENT; MULTIPOINT FLUX APPROXIMATION; QUADRILATERAL MESH;

FINITE ELEMENT METHOD;

EID: 57849167062     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2327     Document Type: Article

References (29)

1. Raviart PA, Thomas JM. A mixed finite element method for second order elliptic problems. Mathematical Aspects of the Finite Element Method. Lectures Notes in Mathematics, vol. 606. Springer: New York, 1977; 292-315.
2. Thomas JM. Sur I'Analyse Numérique des Méthodes d'Eléments Finis Hybrides et Mixtes. Thèse de doctorat d'état, Université Pierre et Marie Curie, France, 1977.
3. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: Berlin, 1991.
4. Chavent G, Jaffré J. Mathematical Models and Finite Elements for Reservoir Simulation. North-Holland: Amsterdam, 1986.
5. Aavatsmark I, Barkve T, Bøe Ø, Marmseth T. Discretization on non-orthogonal, curvilinear grids for multiphase flow. In Proceedings of 4th European Conference on the Mathematics of Oil Recovery, Christie MA, Farmer CL, Guillon O, Heinmann ZE (eds), Norway, 1994.
6. Edwards MG, Rogers CF. A flux continuous scheme for the full tensor pressure equation. Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Røros, 1994.
7. Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids. Journal of Computational Geosciences 2002; 6:404-432.
8. Edwards MG, Rogers CF. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Journal of Computational Geosciences 1998; 2:259-290.
9. Aavatsmark I, Eigestad GT, Klausen RA, Wheeler MF, Yotov I. Convergence of a symmetric MPFA method on quadrilateral grids. Technical Report, Department of Mathematics, University of Pittsburgh, 2005.
10. Edwards MG. Symmetric positive definite general tensor discretization operators on unstructured and flow based grids. Eighth European Conference on Mathematics of Oil Recovery, Freiberg, Germany, 3-6 September 2002.
11. Edwards MG. Control-volume distributed subcell flux schemes for unstructured and flow based grids. SPE Reservoir Simulation Symposium, Houston, TX, U.S.A., 3-5 February 2003.
12. Edwards MG. Unstructured, control-volume distributed, full-tensor finite volume schemes with flow based grids. Computational Geoscience 2002; 6:433-452.
13. Klausen RA, Winther R. Convergence of multipoint flux approximations on quadrilateral grids. Numerical Methods for Partial Differential Equations 2006; 22:1438-1454.
14. Wheeler MF, Yotov I. A cell-centered finite difference on quadrilaterals. Technical Report, Department of Mathematics, University of Pittsburgh, 2005.
15. Wheeler MF, Yotov I. A multipoint flux mixed finite element method. Technical Report, Department of Mathematics, University of Pittsburgh, 2005.
16. Klausen RA, Winther R. Robust convergence of multi-point flux approximations on rough grids. Numerische Mathematik 2006; 3:317-337.
17. Aavatsmark I, Barkve T, Bøe Ø, Mannseth T. Discretization on non-orthogonal, quadrilateral grids for inhomogeneous. anisotropic media. Journal of Computational Physics 1996: 127:2-14.
18. Ewing RE, Liu M, Wang J. Superconvergence of mixed finite element approximation over quadrilaterals. SIAM Journal on Numerical Analysis 1999; 36:772-787.
19. Russell TF, Wheeler MF. Finite element and finite difference methods for continuous flows in porous media. In The Mathematics of Reservoir Simulation, Ewing RE (ed.). Frontiers in Applied Mathematics, vol. 1. SIAM: Philadelphia, 1984; 35-106.
20. Arbogast T, Dawson CN, Keenan PT, Wheeler MF, Yotov I. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM Journal on Scientific Computing 1998; 19(2):404-425.
21. Arbogast T, Wheeler MF, Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM Journal on Numerical Analysis 1997; 34(2):828-852.
22. Younes A, Ackerer P, Mosé R, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. Journal of Computational Physics 1999; 149:148-167.
23. Chavent G, Younes A, Ackerer P. On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles. Computer Methods in Applied Mechanics and Engineering 2003; 192:655-682.
24. Younes A, Ackerer P, Chavent G. From mixed finite elements to finite volumes for elliptic PDE in 2 and 3 dimensions. International Journal for Numerical Methods in Engineering 2004; 59:365-388.
25. Klausen RA, Russell TF. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Computational Geosciences 2004; 8(4):341-377.
26. Pal M, Edwards MG, Lamb AR. Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equation. International Journal for Numerical Methods in Fluids 2006; 51:1177-1203.
27. Eisenstad SC. Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM Journal on Scientific and Statistical Computing 1981; 2:1-4.
28. Davis TA, Duff IS. A combined unifrontal/multifrontal method for unsymmetric sparse matrices. Technical Report TR-97-016, Computer and Information Science and Engineering Department, University of Florida, 1997.
29. Eigestad GT, Klausen RA. On the convergence of the multi-point flux approximation O-method; numerical experiments for discontinuous permeability. Numerical Methods for Partial Differential Equations 2005; 21: 1079-1098.