Colóquio - Multiscale Hybrid-Mixed Methods

Multiscale Hybrid-Mixed Methods
Expositor: Frederic Valentin
Instituição: LNCC
Data e Horário: 16/04/2019 | 16:00 hs

Resumo:  We present an overview and some recent results of a new family of multiscale finite element methods for partial differential equations with highly heterogeneous coefficients, named Multiscale Hybrid-Mixed (MHM) methods. The MHM method consists of a strategy that naturally incorporates multiple scales in the numerical solutions while providing solutions with high-order precision for the primal and dual variables.

It is a consequence of a hybridization procedure, which characterizes the unknowns as a direct sum of a “coarse” solution and the solutions to local problems with boundary conditions driven by the Lagrange multipliers. 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 achieves superconvergence with respect to the mesh parameter and is robust in terms of (small) physical parameters. Also, a face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms.

The general framework is illustrated for the second order elliptic equations, and then further extended to singularly perturbed transport problems, fluid flows and wave propagation models in heterogeneous media. Numerical results verify the theoretical properties, the capacity of the MHM method to accurately incorporate heterogeneity and high-contrast coefficients in the numerical solution, and the great performance of the new a posteriori error estimator in driving mesh adaptativity. We conclude that the MHM method is naturally shaped for parallel computing environments and appears to be a highly competitive option to handle realistic multiscale initial and boundary value problems with precision on coarse meshes.