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In this work we extend the multiscale finite element method (MsFEM)
as formulated by Hou and Wu in [14] to the PDE system of linear elasticity.
The application, motivated from the multiscale analysis of highly heterogeneous
composite materials, is twofold. Resolving the heterogeneities on
the finest scale, we utilize the linear MsFEM basis for the construction of
robust coarse spaces in the context of two-level overlapping Domain Decomposition
preconditioners. We motivate and explain the construction
and present numerical results validating the approach. Under the assumption
that the material jumps are isolated, that is they occur only in the
interior of the coarse grid elements, our experiments show uniform convergence
rates independent of the contrast in the Young's modulus within the
heterogeneous material. Elsewise, if no restrictions on the position of the
high coefficient inclusions are imposed, robustness can not be guaranteed
any more. These results justify expectations to obtain coefficient-explicit
condition number bounds for the PDE system of linear elasticity similar to
existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner
and Scheichl [12]. Furthermore, we numerically observe the properties
of the MsFEM coarse space for linear elasticity in an upscaling framework.
Therefore, we present experimental results showing the approximation errors
of the multiscale coarse space w.r.t. the fine-scale solution.

The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering out solid particles from liquid oil. The Navier-Stokes-Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier-Stokes equations, where the Brinkmann term is considered at both, prediction and correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. Results of computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.

In this paper we consider numerical algorithms for solving a system of nonlinear PDEs arising in modeling of liquid polymer injection. We investigate the particular case when a porous preform is located within the mould, so that the liquid polymer flows through a porous medium during the filling stage. The nonlinearity of the governing system of PDEs is due to the non-Newtonian behavior of the polymer, as well as due to the moving free boundary. The latter is related to the penetration front and a Stefan type problem is formulated to account for it. A finite-volume method is used to approximate the given differential problem. Results of numerical experiments are presented. We also solve an inverse problem and present algorithms for the determination of the absolute preform permeability coefficient in the case when the velocity of the penetration front is known from measurements. In both cases (direct and inverse problems) we emphasize on the specifics related to the non-Newtonian behavior of the polymer. For completeness, we discuss also the Newtonian case. Results of some experimental measurements are presented and discussed.

On a Multigrid Adaptive Refinement Solver for Saturated Non-Newtonian Flow in Porous Media A multigrid adaptive refinement algorithm for non-Newtonian flow in porous media is presented. The saturated flow of a non-Newtonian fluid is described by the continuity equation and the generalized Darcy law. The resulting second order nonlinear elliptic equation is discretized by a finite volume method on a cell-centered grid. A nonlinear full-multigrid, full-approximation-storage algorithm is implemented. As a smoother, a single grid solver based on Picard linearization and Gauss-Seidel relaxation is used. Further, a local refinement multigrid algorithm on a composite grid is developed. A residual based error indicator is used in the adaptive refinement criterion. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Several results from numerical experiments are presented in order to examine the performance of the solver.

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Efficient solution of the resulting second order nonlinear elliptic equation is discussed here. The equation is discretized by a finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here, as an essential part of the accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid - adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.

In this paper, we propose multi-level Monte Carlo(MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multi-phase flow and transport. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. We consider two types of ensemble level mixed multiscale finite element methods, (1) the no-local-solve-online ensemble level method (NLSO) and (2) the local-solve-online ensemble level method (LSO). Both mixed multiscale methods use a number of snapshots of the permeability media to generate a multiscale basis.
As a result, in the offline stage, we construct multiple basis functions for
each coarse region where basis functions correspond to different realizations.
In the no-local-solve-online ensemble level method one uses the whole set of pre-computed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method one uses the pre-computed functions to construct a multiscale basis for a particular realization. With this basis the solution corresponding to this
particular realization is approximated in LSO mixed MsFEM. In both approaches
the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to pre-compute a
multiscale basis. We note that LSO approaches share similarities with reduced basis methods [11, 21, 22].
In multi-level Monte Carlo methods ([14, 13]), more accurate (and expensive) forward simulations are run with fewer samples while less accurate(and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations carefully, one can show that MLMC methods can provide better accuracy
at the same cost as MC methods. In our simulations, our goal is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. In particular, we show that NLSO
mixed MsFEM is more accurate compared to LSO mixed MsFEM. Further, we use both approaches in the context of MLMC to speed-up MC
calculations. We present basic aspects of the algorithm and numerical
results for coupled flow and transport in heterogeneous porous media.

Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of convergence 1.5 is proven for arbitrary located interface, and second order convergence is proven for the case when the interface coincides with a grid node. Furthermore, following the ideas from [3], modified HA discretization are suggested for particular cases. The velocity and the stress are approximated with second order on the interface in this case. It is shown that for wide class of problems, the modified discretization provides better accuracy. Second order convergence for modified scheme is proven for the case when the interface coincides with a displacement grid node. Numerical experiments are presented in order to illustrate our considerations.

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.

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.