Mixed multiscale finite element methods using approximate global information based on partial upscaling
KAUST Grant NumberKUS-CI-016-04
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AbstractThe use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308-317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods. © 2009 Springer Science+Business Media B.V.
CitationJiang L, Efendiev Y, Mishev Ii (2009) Mixed multiscale finite element methods using approximate global information based on partial upscaling. Computational Geosciences 14: 319–341. Available: http://dx.doi.org/10.1007/s10596-009-9165-7.
Sponsorscomments and suggestions to improve the paper. We are also grateful to Jorg Aarnes for providing some mixed MsFEM codes. L. Jiang would like to acknowledge partial support from Exxonmobil URC and Chinese NSF 10901050. Y. Efendiev would like to acknowledge partial support from NSF and DOE. Y. Efendiev's work was also partially supported by Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST).