Non-overlapping domain decomposition multiscale methods have been succesfully applied to flows in porous media. Such remarkable class of methods seek to decompose the domain of the porous media flow equations in non-overlapping subdomains, solving smaller local problems in parallel, and one global interface problem, instead of a large coupled one. Usually the interface problem enforces the compatibility conditions — continuity of pressures and normal fluxes — across subdomain interfaces. In multiscale methods, these conditions are enforced at different length scales, originating different methods. Examples are the MMMFEM  (Multiscale Mixed Mortar Finite Element Method), that prioritize pressure continuity at fine scales and weak flux continuity; the MHM  (Multiscale Hybrid-Mixed Method), that enforces continuity of normal fluxes at fine scales and weak pressure continuity; the MuMM  (Multiscale Mixed Method) that enforces weak continuity of both pressure and normal fluxes through Robin-type boundary conditions. Recently, the MRCM  (Multiscale Robin Coupled Method) that generalizes the aforementioned methods in one single variational formulation has been introduced.
It is well known that enforcing continuity only at larger scales creates a discrepancy in the continuity (of pressure or normal fluxes, or both) in fine scales, so that a downscaling is required to keep the resulting velocity fields conservative.
We propose a new iterative procedure based on alternating domains with minimum overlapping to perform downscaling of the computed normal fluxes, resulting in new conservative velocity fields. We studied the applicability and efficiency of this new method when applied to multiphase flow problems using the MRCM, as compared to existing techniques, illustrating the advantages of our new procedure.
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