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On the relationship between the multiscale finite volume method and domain decomposition preconditioners. (English) Zbl 1155.76042
Summary: We review the classical nonoverlapping domain decomposition (NODD) preconditioners, together with the newly developed multiscale control volume (MSCV) method. By comparing the formulations, we observe that the MSCV method is a special case of a NODD preconditioner. We go on to suggest how the more general framework of NODD can be applied in the multiscale context to obtain improved multiscale estimates.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
Full Text: DOI
[1] Aarnes, J.E., Kippe, V., Lie, K.A.: Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour. 28(3), 257–271 (2005)
[2] Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002) · Zbl 1094.76550
[3] Arbogast, T.: Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42(2), 576–598 (2004) · Zbl 1078.65092
[4] Barrett, R. et al.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM (1994)
[5] Bjørstad, P.E., Widlund, O.: Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23(6), 1093–1120 (1986) · Zbl 0615.65113
[6] Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. J. Numer. Anal. 47, 103–134 (1986) · Zbl 0615.65112
[7] Cai, X.C., Gropp, W.D., Keyes, D.E.: A comparison of some domain decomposition algorithms for nonsymmeric elliptic problems. In: Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 224–235 (1992) · Zbl 0770.65079
[8] Chan, T.F., Keyes, D.E.: Interface preconditioning for domain-decomposed convection-diffusion operators. In: Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 245–262 (1990) · Zbl 0722.65060
[9] Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26(10), 1041–1060 (2003)
[10] Efendiev, Y.R., Durlofsky, L.J.: Accurate subgrid models for two-phase flow in heterogeneous reservoirs. SPE J. 9(2), 219–226 (2004)
[11] Espedal, M.S., Ewing, R.E.: Characteristic Petrov-Galerkin subdomain methods for 2-phase immiscible flow. Comput. Methods Appl. Mech. Eng. 64(1–3), 113–135 (1987) · Zbl 0607.76103
[12] Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003) · Zbl 1047.76538
[13] Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul. 3(1), 50–64 (2004) · Zbl 1160.76372
[14] Smith, B.F.: An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems. SIAM J. Sci. Stat. Comput. 13(1), 364–378 (1992) · Zbl 0751.73059
[15] Smith, B.F., Bjørstad, P.E., Gropp, W.D.: Domain Decomposition. Cambridge University Press, Cambridge (1996) · Zbl 0857.65126
[16] Toselli, A., Widlund, O.: Domain decomposition methods–algorithms and theory. Springer Ser. Comput. Math. 34 (2005) · Zbl 1069.65138
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