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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.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[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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