The multiscale restricted-smoothing basis (MsRSB) method is the current state-of-the-art within multiscale methods. MsRSB is very robust and versatile can either be used as an approximate coarse-scale solver having mass-conservative subscale resolution, or as an iterative fine-scale solver that will provide mass-conservative solutions for any given tolerance. The performance of the method has been demonstrated on incompressible 2-phase flow, compressible 2 and 3-phase black oil models, as well as compositional models. It has also been demonstrated that the method can utilize combinations of multiple prolongation operators, e.g., corresponding to coarse grids withdifferent resolutions, adapting to geological features, adapting to wells, or moving displacement fronts.

This chapter explains the basic ideas of the MsRSB method, including methods to construct coarse partitions, prolongation, and restriction operators, reduction of the fine-scale flow equations to a coarse-scale system, and formulation as part of a two-level iterative solver. We outline the key functions in the module and  show various examples of how the method can be used as an iterative solver for incompressible and compressible flow on 2D rectilinear grids, unstructured grids, and 3D stratigraphic grids.

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