Latest developments on the Multiscale Robin Coupled Method for flows in heterogeneous porous media
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The Multiscale Robin Couple Method (MRCM) [1] is a multiscale domain decomposition method based on a mixed finite element formulation that allows for efficient solutions of second-order elliptic equations in a coarse scale, incorporating fine grid details of the solution through the efficient parallel computation of independent multiscale basis functions. The coupling between subdomains is performed by the imposition of Robin-type boundary conditions in the computation of the multiscale basis functions, ensuring that compatibility conditions are enforced on a large scale. This procedure generalizes well-known multiscale mixed methods, such as the Multiscale Mortar Mixed Finite Element Method (MMMFEM) [2], the Multiscale Hybrid-Mixed Method (MHM) [3], and the Multiscale Mixed Method (MuMM) [4], via the suitable choice of the Robin boundary parameter and interface spaces. While generalizing those methods, it also introduces the possibility of adaptivity, resulting in accurate solutions compared to the undecomposed fine grid solution and other multiscale procedures based on the lowest order Raviart-Thomas finite element spaces. We will present the latest developments of the MRCM, in terms of interface enrichment and adaptivity, with a focus on the accuracy of the decomposition, applications to complex porous media flow models, preconditioning, and speedup results on high-performance computations involving billions of fine grid cells.
