Recent results on the multiscale hybrid-mixed method for Stokes and Brinkman equations
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The Multiscale Hybrid-Mixed method (MHM for short) is a consequence of a continuum-level hybridization procedure that characterizes the unknowns as a direct sum of a ``coarse'' solution and the solutions to problems with Neumann boundary conditions driven by the multipliers. As a result, the MHM method becomes a strategy that naturally incorporates multiple scales through multiscale basis functions, while providing high-order accurate solutions for both primal and dual variables. The completely independent local problems are incorporated into the upscaling procedure, and then computational approximations can be obtained naturally in a parallel computing environment. This work presents recent results for a family of MHM methods for Stokes and Brinkman flows. Numerical analysis for the one- and two-level versions of the MHM method with stabilized and stable local solvers shows that they are optimally convergent and can achieve superconvergence with locally conservative velocity fields. The methodology is illustrated using several numerical tests including a highly heterogeneous coefficient problem.