A note on variational multiscale methods for high-contrast heterogeneous porous media flows with rough source terms

Advances in Water Resources

Volumn 34, Issue 9, 2011, Pages 1177-1185

  Calo, Victor ^a   Efendiev, Yalchin ^b   Galvis, Juan ^b  

^a KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Heterogeneous; High contrast; Multiscale; Multiscale finite element; Source; Variational multiscale

Indexed keywords

HETEROGENEOUS; HIGH CONTRAST; MULTISCALE FINITE ELEMENT; MULTISCALES; SOURCE; VARIATIONAL MULTISCALE;

POROUS MATERIALS;

CONTRAST MEDIA;

ERROR CORRECTION; FINITE ELEMENT METHOD; FLUID FLOW; PERMEABILITY; POROUS MEDIUM; TESTING METHOD; VARIANCE ANALYSIS;

EID: 80052868267     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2010.12.011     Document Type: Article

References (35)

1. Aarnes J.E. On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. SIAM MMS 2004, 2:421-439.
2. Aarnes J.E., Efendiev Y., Jiang L. Analysis of multiscale finite element methods using global information for two-phase flow simulations 2008, SIAM MMS.
3. Aarnes J., Hou T. Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Math Appl Sin Engl Ser 2002, 18(1):63-76.
4. Aarnes J.E., Krogstad S., Lie K.-A. A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform grids. Multiscale Model Simul 2006, 5(2):337-363.
5. Arbogast T. Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput Geosci 2002, 6:453-481.
6. Arbogast T., Boyd K.J. Subgrid upscaling and mixed multiscale finite elements. SIAM J Numer Anal 2006, 44(3):1150-1171.
7. Arbogast T., Pencheva G., Wheeler M.F., Yotov I. A multiscale mortar mixed finite element method. SIAM J Multiscale Model Simul 2007, 6(1):319-346.
8. Bazilevs Y., Calo V.M., Cottrell J.A., Hughes T.J.R., Reali A., Scovazzi G. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 2007, 197(1-4):173-201.
9. Berlyand L, Owhadi H. A new approach to homogenization with arbitrary rough high contrast coefficients for scalar and vectorial problems, submitted for publication.
10. Chen Z., Hou T.Y. A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math Comput 2002, 72:541-576.
11. Chu C.C., Graham I.G., Hou T.Y. A new multiscale finite element methods for high-contrast elliptic interface problem. Math Comput 2010, 79:1915-1955.
12. Efendiev Y., Galvis J., Wu X.H. Multiscale finite element methods for high-contrast problems using local spectral basis functions. J Comput Phys 2011, 230(4):937-955.
13. Efendiev Y., Ginting V., Hou T., Ewing R. Accurate multiscale finite element methods for two-phase flow simulations. J Comput Phys 2006, 220(1):155-174.
14. Efendiev Y., Hou T. Multiscale finite element methods. Theory and applications 2009, Springer.
15. Galvis J., Efendiev Y. Domain decomposition preconditioners for multiscale flows in high-contrast media: reduced dimension coarse spaces. SIAM MMS 2010, 8(5):1621-1644.
16. Galvis J., Efendiev Y. Domain decomposition preconditioners for multiscale flows in high contrast media. SIAM MMS 2010, 8(4):1461-1483.
17. Graham I.G., Lechner P.O., Scheichl R. Domain decomposition for multiscale PDEs. Numer Math 2007, 106(4):589-626.
18. Graham I.G., Scheichl R. Robust domain decomposition algorithms for multiscale PDEs. Numer Methods Partial Differ Equat 2007, 23(4):859-878.
19. Hackbush W. Local defect correction and domain decomposition techniques. Computing Supplementary 1983, vol. 5:89-113. Springer, Wien, New York. K. Bohmer, H.J. Stetter (Eds.).
20. Hajibeygi H., Bonfigli G., Hesse M., Jenny P. Iterative multiscale finite-volume method. J Comput Phys 2008, 227(19):8604-8621.
21. Hou T.Y., Wu X.H. A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 1997, 134:169-189.
22. Hsu M.C., Bazilevs Y., Calo V.M., Tezduyar T.E., Hughes T.J.R. Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Meth Appl Mech Engrg 2010, 199(13-16):828-840.
23. Hughes T.J.R. Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Meth Appl Mech Engrg 1995, 127(1-4):387-401.
24. Hughes T.J.R., Feijoo G., Mazzei L., Quincy J.B. The variational multiscale method - a paradigm for computational mechanics. Comput Meth Appl Mech Engrg 1998, 166(1-2):3-24.
25. Jenny P., Lee S.H., Tchelepi H. Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J Comput Phys 2003, 187:47-67.
26. Jenny P., Lunati I. Modeling complex wells with the multi-scale finite-volume method. J Comput Phys 2009, 228(3):687-702.
27. Lunati I, Jenny P. Multi-scale finite-volume method for highly heterogeneous porous media with shale layers. In: Proceedings of the 9th European conference on the mathematics of oil recovery (ECMOR), Cannes, France, 2004.
28. Lunati I., Lee S.H. An operator formulation of the multiscale finite-volume method with correction function. Multiscale Model Simul 2009, 8(1):96-109.
29. Mathew T.P.A. Domain decomposition methods for the numerical solution of partial differential equations. Lecture notes in computational science and engineering 2008, vol. 61. Springer-Verlag, Berlin.
30. Nordbotten J.M., Bjorstad P.E. On the relationship between the multiscale finite-volume method and domain decomposition preconditioners. Comput Geosci 2008, 12(3):367-376.
31. Owhadi H., Zhang L. Metric based up-scaling. Commun Pure Appl Math, Commun Pure Appl Math LX 2007, 675-723.
32. Tosseli A., Widlund O. Domain decomposition methods - algorithms and theory. Springer series in computational mathematics 2005, vol. 34. Springer-Verlag, Berlin.
33. Vassilevski P.S. Multilevel block factorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations 2008, Springer.
34. Xu J., Zikanov L. On an energy minimizing basis for algebraic multigrid methods. Comput Visual Sci 2004, 7:121-127.
35. Xu J., Zou J. Some nonoverlapping domain decomposition methods. SIAM Rev 1998, 40(4):857-914.