Abstract. 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.Mathematics Subject Classification. 65F10, 65N20, 65N22, 65N30, 65N55.

Abstract. We present a two-scale finite element method for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.

A general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term "robust" refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of (nonlinear) algebraic multilevel iterations. The crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems. The method is analyzed in a rather general setting applying to the scalar elliptic equation, the equations of linear elasticity, and equations arising in the solution of Maxwell's equations. Numerical results for the scalar elliptic equation are presented showing its robust convergence behavior.1. Introduction. The discretization of many important partial differential equations (PDEs) leads to symmetric positive definite (SPD) systems. Important instances of such PDEs are the stationary heat equation, the equations of linear elasticity, and equations arising in the solution of Maxwell's equations. Since in many applications these systems can be (very) large, one is interested in designing solution schemes whose convergence rates are independent of the problem sizes, i.e., mesh parameters. Furthermore, in numerous practically important situations one deals with highly varying or otherwise degenerate coefficients accounting for the physical properties of the underlying process. Examples of such coefficients can be observed when modeling porous media flow in domains with highly varying permeability fields. Other degenerate coefficients arise, e.g., in linear elasticity when considering (even homogeneous) almost incompressible media. Overall, there are many problems for which it is desirable to construct efficient solution schemes whose convergence rates are robust with respect to the problem size as well as problem parameters.The issue of achieving robustness with respect to mesh parameters has been successfully addressed by several iterative solution schemes. Here we in particular mention domain decomposition (DD) (see, e.g., [18,24]) and multilevel/multigrid methods (see, e.g., [25,27]). Achieving robustness with respect to problem parameters has proved to be a more difficult task. For standard two-level DD methods applied to the scalar elliptic equation with varying coefficient it has been shown (see [12,24]) that robust convergence rates are obtained provided that the variations of the coefficients inside coarse grid cells are bounded.

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 numerical method for computing 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 method 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. The applicability of the presented approach to upscaling other similar problems, e.g. flow in fractured porous media, is also discussed.

We are interested in numerical methods for computing the effective heat conductivities of fibrous insulation materials, such as glass or mineral wool, characterized by low solid volume fractions and high contrasts, i.e., high ratios between the thermal conductivities of the fibers and the surrounding air. We consider a fast numerical method for solving some auxiliary cell problems appearing in this upscaling procedure. The auxiliary problems are boundary value problems of the steady-state heat equation in a representative elementary volume occupied by fibers and air. We make a simplification by replacing these problems with appropriate boundary value problems in the domain occupied by the fibers only. Finally, the obtained problems are further simplified by taking advantage of the slender shape of the fibers and assuming that they form a network. A discretization on the graph defined by the fibers is presented and error estimates are provided. The resulting algorithm is discussed and the accuracy and the performance of the method are illusrated on a number of numerical experiments