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The discrete system that arises from this mimetic FDM is of the same form as (20). The only difference at the discrete level is that the entries in B and g are computed using the m(·, ·) inner-product instead of the L2 inner-product (u, λ −1 v) on H div (Ω ). Thus, for the mimetic FDM we have gkl = m(ρΞ , ψkl ) , where Ξ = ∑i j ξi j ψi j and ξi j = 1 |γi j | γi j bi j,kl = m(ψi j , ψkl ) , G · ni j ds. 22 J. E. –A. Lie, V. Kippe, S. Krogstad Fig. 7. Examples of deformed and degenerate hexahedral cells arising in corner-point grid models.

Sci. , 19(5):1717–1736, 1998. 8. T. Arbogast. Numerical subgrid upscaling of two-phase flow in porous media. In Z. Chen, R. E. -C. , pages 35–49. Springer-Verlag, Berlin, 2000. 9. T. Arbogast. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. , 42(2):576–598, 2004. 10. K. Aziz and A. Settari. Petroleum reservoir simulation. Elsevier, London and New York, 1979. 11. I. Babu˘ska, G. Caloz, and E. Osborn. Special finite element methods for a class of second order elliptic problems with rough coefficients.

The MsFEM was first introduced by Hou and Wu [33], but the basic idea goes back to earlier work by Babu˘ska and Osborn [12] for 1D problems and Babu˘ska, Caloz, and Osborn [11] for special 2D problems. The method is, like standard FEMs, based on a variational formulation. In the variational formulation of (27) we seek p ∈ H01 (Ω ) such that a(p, v) = ( f , v) for all v ∈ H01 (Ω ), where (·, ·) is the L2 inner-product and a(p, v) = Ω K(x)u (x)v (x) dx. (28) 30 J. E. –A. Lie, V. Kippe, S. Krogstad Now, let NB = {0 = x0 < x1 < .

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