HPC for Energy (HPC4E) researcher Frédéric Valentin (LNCC) presented the talk 'Multiscale Hybrid-Mixed Methods for Fluids' at the second Mathematical Congress of the Americas (MCA2017), held on the 24 to 28 July in Montréal (Canada). The talk was performed in the session on 'Applied Math and Computational Science across the Americas'.
This work presents a family of finite element methods for multiscale fluid problems, named Multiscale Hybrid-Mixed (MHM) methods. The MHM method is a consequence of a hybridization procedure which 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 while providing solutions with high-order precision for the primal and dual variables. The completely independent local problems are embedded in the upscaling procedure, and computational approximations may be naturally obtained in a parallel computing environment. Well-posedness and best approximation results for the one- and two-level versions of the MHM method show that the method is optimal convergent and achieves super-convergence for the velocity field with respect to the mesh parameter. Interesting, the numerical velocity field turns out to be locally conservative. Also, a face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms. The general framework and some recent results are illustrated for the Stokes and Brinkman equations, and validated through a large varieties of numerical results for highly heterogeneous coefficient problems.
As explained in the website, the goal of the Mathematical Congress of the Americas is to internationally highlight the excellence of mathematical achievements in the Americas and foster collaborations among researchers, students, institutions and mathematical societies in the Americas. The first MCA was held in Mexico in 2013. This inaugural congress garnered the participation of about 1,000 mathematicians and students from all over the Americas and other parts of the world.
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