ESAIM: Mathematical Modelling and Numerical Analysis, 2012
An abstract framework for constructing stable decompositions of the spaces corresponding to gener... more An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461-1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
An efficient approach for calculating the effective heat conductivity for a class of industrial c... more 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 δ of low and high conductivities satisfies δ 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(δ) 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.
An abstract framework for constructing stable decompositions of the spaces corresponding to gener... more An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis, Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
... Some preliminary results have been reported in [2]. References [1] T. Arbogast, Analysis of a... more ... Some preliminary results have been reported in [2]. References [1] T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal., 42 (2) (2004), 576598. [2] OP Iliev, RD Lazarov, and J. Willems, Discontinuous Galerkin ...
ESAIM: Mathematical Modelling and Numerical Analysis, 2012
An abstract framework for constructing stable decompositions of the spaces corresponding to gener... more An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461-1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
An efficient approach for calculating the effective heat conductivity for a class of industrial c... more 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 δ of low and high conductivities satisfies δ 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(δ) 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.
An abstract framework for constructing stable decompositions of the spaces corresponding to gener... more An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis, Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
... Some preliminary results have been reported in [2]. References [1] T. Arbogast, Analysis of a... more ... Some preliminary results have been reported in [2]. References [1] T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal., 42 (2) (2004), 576598. [2] OP Iliev, RD Lazarov, and J. Willems, Discontinuous Galerkin ...
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