Free Access
Volume 46, Number 4, July-August 2012
Page(s) 759 - 796
Published online 03 February 2012
  1. J.E. Aarnes, S. Krogstad and K.-A. Lie, A hierarchical multiscale method for two-phase flow based on mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5 (2006) 337–363. [CrossRef] [MathSciNet] [PubMed]
  2. J.E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information. Multiscale Model. Simul. 7 (2008) 655–676. [CrossRef] [MathSciNet] [PubMed]
  3. I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405–432. [CrossRef] [MathSciNet] [PubMed]
  4. I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. [CrossRef] [MathSciNet]
  5. I. Aavastsmark, G.T. Eigestad, R.A. Klausen, M.F. Wheeler and I. Yotov, Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333–345. [CrossRef] [MathSciNet] [PubMed]
  6. I. Aavatsmark, G.T. Elgestad, B.T. Mallison and J.M. Nordbotten, A compact multipoint flux approximation method with improved robustness. Numer. Methods for Partial Differential Equations 24 (2008) 1329–1360. [CrossRef] [MathSciNet]
  7. L. Agélas, D.A. Di Pietro and J. Droniou, The G method for heterogeneous anisotropic diffusion on general meshes. Math. Model. Numer. Anal. 44 (2010) 597–625. [CrossRef] [EDP Sciences] [MathSciNet]
  8. T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM. J. Numer. Anal. 42 (2004) 576–598. [CrossRef] [MathSciNet]
  9. T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. [CrossRef] [MathSciNet]
  10. T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19 (1998) 404–425. [CrossRef] [MathSciNet]
  11. 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) 1295–1315. [CrossRef] [MathSciNet]
  12. T. Arbogast, G. Pencheva, M.F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6 (2007) 319–346. [CrossRef] [MathSciNet] [PubMed]
  13. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19 (1985) 7–32. [MathSciNet]
  14. D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429–2451. [CrossRef] [MathSciNet]
  15. J. Baranger, J.F. Maitre and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modèl. Math. Anal. Numèr. 30 (1996) 445–465. [CrossRef] [EDP Sciences] [MathSciNet]
  16. C. Bernardi, Y. Maday and A.T. Patera. A new nonconforming approach to domain decomposition : The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, edited by H. Brezis and J.L. Lions. Longman Scientific and Technical, Harlow, UK (1994).
  17. M. Berndt, K. Lipnikov, M. Shashkov, M.F. Wheeler and I. Yotov, Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM. J. Numer. Anal. 43 (2005) 1728–1749. [CrossRef] [MathSciNet]
  18. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Springer-Verlag (2007).
  19. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).
  20. F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. [CrossRef] [MathSciNet]
  21. F. Brezzi, J. Douglas, R. Duran and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. [CrossRef] [MathSciNet]
  22. F. Brezzi, M. Fortin and L.D. Marini, Error analysis of piecewise constant pressure approximations of Darcy’s law. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1547–1559. [CrossRef] [MathSciNet]
  23. Z. Cai, J.E. Jones, S.F. McCormick and T.F. Russell, Control-volume mixed finite element methods. Comput. Geosci. 1 (1997) 289–315 (1998). [CrossRef] [MathSciNet] [PubMed]
  24. Z. Chen and T.Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72 (2003) 541–576. [CrossRef] [MathSciNet]
  25. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4. North-Holland, Amsterdam (1978); reprinted, SIAM, Philadelphia (2002).
  26. R. Duran, Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990) 287–298. [CrossRef] [MathSciNet]
  27. M.G. Edwards, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002) 433–452. [CrossRef] [MathSciNet] [PubMed]
  28. M.G. Edwards and C.F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259–290 (1999). [CrossRef] [MathSciNet] [PubMed]
  29. R.E. Ewing, M.M. Liu and J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals. SIAM. J. Numer. Anal. 36 (1999) 772–787. [CrossRef] [MathSciNet]
  30. R. Eymard, T. Gallouet and R. Herbin, Finite volume methods. in Handbook of Numerical Analysis. North-Holland, Amsterdam (2000) 713–1020.
  31. G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I. Linearized steady problems, Springer-Verlag, New York (1994)
  32. B. Ganis and I. Yotov, Implementation of a mortar mixed finite element using a multiscale flux basis. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3989–3998. [CrossRef] [MathSciNet]
  33. V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).
  34. R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, edited by R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux. SIAM, Philadelphia (1988) 144–172.
  35. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1995).
  36. T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. [CrossRef] [MathSciNet]
  37. T.J.R. Hughes, G.R. Feijoo, L. Mazzei and J.-B. Quincy, The variational multiscale method–a paradim for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. [CrossRef] [MathSciNet]
  38. J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problem in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130–148. [CrossRef] [MathSciNet]
  39. R. Ingram, M.F. Wheeler and I. Yotov, A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48 (2010) 1281–1312. [CrossRef] [MathSciNet] [PubMed]
  40. P. Jenny, S.H. Lee and H.A. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47–67. [CrossRef]
  41. R.A. Klausen and R. Winther, Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104 (2006) 317–337. [CrossRef] [MathSciNet]
  42. R.A. Klausen and R. Winther, Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differential Equations 22 (2006) 1438–1454. [CrossRef] [MathSciNet]
  43. J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, Heidelberg, New York (1972).
  44. K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115–152. [CrossRef] [MathSciNet]
  45. T.P. Mathew, Domain Decomposition and Iterative Methods for Mixed Finite Element Discretizations of Elliptic Problems. Tech. Report 463, Courant Institute of Mathematical Sciences, New York University, New York (1989).
  46. J.C. Nedelec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. [CrossRef] [MathSciNet]
  47. G. Pencheva and I. Yotov, Balancing domain decomposition for mortar mixed finite element methods on non-matching grids. Numer. Linear Algebra Appl. 10 (2003) 159–180. [CrossRef] [MathSciNet]
  48. P.A. Raviart and J. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Mathematical aspects of the Finite Elements Method, Lect. Notes Math. 606 (1977) 292–315. [CrossRef]
  49. J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis II, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science Publishers B.V. (1991) 523–639.
  50. T.F. Russell and M.F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, in The Mathematics of Reservoir Simulation, edited by R.E. Ewing. SIAM, Philadelphia (1983) 35–106.
  51. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. [CrossRef] [MathSciNet] [PubMed]
  52. J.M. Thomas, These de Doctorat d’etat, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Ph.D. thesis, à l’Université Pierre et Marie Curie (1977).
  53. M. Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM :M2AN 40 (2006) 367–391. [CrossRef] [EDP Sciences] [MathSciNet]
  54. J. Wang and T.P. Mathew, Mixed finite element method over quadrilaterals, in Conference on Advances in Numerical Methods and Applications, edited by I.T. Dimov, B. Sendov and P. Vassilevski. World Scientific, River Edge, NJ (1994) 351–375.
  55. A. Weiser and M.F. Wheeler, On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351–375. [CrossRef] [MathSciNet] [PubMed]
  56. M.F. Wheeler and I. Yotov, A multipoint flux mixed finite element method. SIAM. J. Numer. Anal. 44 (2006) 2082–2106. [CrossRef] [MathSciNet] [PubMed]
  57. M.F. Wheeler, G. Xue and I. Yotov, A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Accepted by Numer. Math. (2011).
  58. A. Younès, P. Ackerer and G. Chavent, From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg. 59 (2004) 365–388. [CrossRef] [MathSciNet]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you