Ensemble level multiscale finite element and preconditioner for channelized systems and applications

Abstract

We consider multiscale flow in porous media. We assume that we can characterize the ensemble of all possible flow scenarios, that is, we can describe all possible permeability configurations needed for the simulations. We construct coarse basis functions that can provide inexpensive coarse approximations that are: (1) adequate for all possible flow scenarios in the given ensemble, (2) robust with respect to the small scales and high variations in each flow scenario. The coarse approximations developed here can be used as a multiscale finite element method, or as the coarse solver in a two-level domain decomposition iterative method. The methods presented here extend, to the ensemble case, some of the results in [J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media: reduced dimension coarse spaces, SIAM Multiscale Model. Simul. 8 (5) (2010) 1621–1644] and [Y. Efendiev, J. Galvis, X.H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys. 230 (4) (2011) 937–955]. Specifically, ensembles of permeability fields with high-contrast channels and inclusions are considered. Our main objective here is to construct special multiscale basis functions for the whole ensemble of flow scenarios. The coarse basis functions are pre-computed for (selected or constructed) permeability fields with certain topological properties. This procedure is a preprocessing step and it avoids constructing basis functions (or computing upscaling parameters) for each permeability realization. Then, for any permeability, the solution of elliptic equation can be project to the space spanned by these pre-computed basis functions. We apply this coarse multiscale solver to the design of two-level domain decomposition preconditioner. Numerical experiments show that the ensemble level multiscale finite element method converges to the reference solution. Numerical experiments also show that the ensemble level domain decomposition preconditioner condition number is independent of the high-contrast in the coefficient.