Frédéric Valentin (LNCC) was invited as a speaker in the XXXVI Congresso Nacional de Matemática Aplicada e Computacional (CNMAC) in Brazil and presented the research carried out in the framework of the HPC4E project: "A Multiscale Hybrid-Mixed Method for the Stokes and Brinkman Equations". He presented this research in the Minisymposium MS6: Novos Desafios na Simulação Numérica de EDPs – II.
This work presents an overview of a family of finite element methods for multiscale problems, named Multiscale Hybrid-Mixed (MHM) methods. MHM methods are a consequence of a hybridization procedure which characterize 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. Also interesting is that the dual variable preserves the local conservation property using a simple post-processing of the primal variable.
Well-posedness and best approximation results for the one- and two-level versions of the MHM method show that the method achieves super-convergence 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 MHM method, along with its associated a posteriori estimator, is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale boundary value problems with precision on coarse meshes.
The general framework and some recent results are illustrated for fluid flow models (Darcy and Stokes equations) and solid model (linear elasticity equation), a singularly perturbed transport problem (reactive-advective dominated equation), and a wave propagation problem (Maxwell equation). Particularly, we highlight how these MHM methods can be derived and analyzed within a common abstract setting, and we show a large varieties of numerical results for highly heterogeneous coefficient problems.
Please, feel free to download the research report from here (PDF).