Mixed Multiscale Finite Element Methods Using Limited Global Information

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Abstract
In this paper, we present a mixed multiscale finite element method using limited global information. We consider a general case where multiple global information is given such that the solution depends smoothly on these global fields. The global fields typically contain small-scale (local or global) information required for achieving a convergence with respect to the coarse mesh size. We present a rigorous analysis and show that the proposed mixed multiscale finite element methods converge. Some preliminary numerical results are shown. We study a parameter dependent permeability field (a simplified case for general stochastic permeability fields). As for spatial heterogeneities, channelized permeability fields with strong nonlocal effects are considered. Using a few global fields corresponding to realizations of permeability fields, we show that one can achieve high accuracy in numerical simulations.