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#### Keywords

- numerical upscaling (2)
- Darcy’s law (1)
- Multiscale problem (1)
- effective heat conductivity (1)
- fibrous insulation materials (1)
- metal foams (1)
- multigrid (1)
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- permeability of fractured porous media (1)
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An efficient approach for calculating the effective heat conductivity for a class of industrial composite materials, such as metal foams, fibrous glass materials, and the like, is discussed. These materials, used in insulation or in advanced heat exchangers, are characterized by a low volume fraction of the highly conductive material (glass or metal) having a complex, network-like structure and by a large volume fraction of the insulator (air). We assume that the composite materials have constant macroscopic thermal conductivity tensors, which in principle can be obtained by standard up-scaling techniques, that use the concept of representative elementary volumes (REV), i.e. the effective heat conductivities of composite media can be computed by post-processing the solutions of some special cell problems for REVs. We propose, theoretically justify, and numerically study an efficient approach for calculating the effective conductivity for media for which the ratio of low and high conductivities satisfies 1. In this case one essentially only needs to solve the heat equation in the region occupied by the highly conductive media. For a class of problems we show, that under certain conditions on the microscale geometry, the proposed approach produces an upscaled conductivity that is O() close to the exact upscaled permeability. A number of numerical experiments are presented in order to illustrate the accuracy and the limitations of the proposed method. Applicability of the presented approach to upscaling other similar problems, e.g. flow in fractured porous media, is also discussed.

Two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media is presented. Accurate finite volume discretization based on multipoint flux approximation (MPFA) for 3D pressure equation is employed to account for the jump discontinuities of full permeability tensors. DD/MG type preconditioner for above mentioned problem is developed. Coarse scale operator is obtained from a homogenization type procedure. The influence of the overlapping as well as the influence of the smoother and cell problem formulation is studied. Results from numerical experiments are presented and discussed.

Abstract — Various advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.