Objectifs :
La formation se déroulera en anglais.
Multiscale finite element methods approach the solution of partial differential equations with heterogeneous coefficients on coarse partitions, and are becoming an attractive alternative to classical finite element methods that require very fine meshes. This five-lecture course will present an overview of the origin of the Multiscale Hybrid-Mixed method (MHM for short) as well as the MHM's basic theory and practical aspects of the implementation of the MHM method. The MHM is a byproduct of a general methodology that uses the hybridization procedure on the continuous level to characterize the unknowns as a direct sum of a ``coarse'' global solution and the solutions of local Neumann problems. At the discrete level, the local problems drive the multiscale basis functions, while the global ``coarse'' problem is responsible for the computation of the (face) degrees of freedom. Since the local problems are entirely independent, the method takes full advantage of parallel computations. The MHM method has exciting features such as super-convergence, robustness in terms of (small) physical parameters, and local conservation properties on general polytopal meshes. Also, it provides a face-based a posteriori error estimator which drives adaptative strategies. Those combined features make the method an affordable and highly precise strategy to solve massive multiscale problems in complicated domains. The MHM method also shares similarities (and differences) with other multiscale and domain decomposition methods. All those aspects will be illustrated during this course through numerical simulations using the MultiScale FEM Library (MSL) from the IPES Research Group (http://ipes.lncc.br/) at the National Laboratory for Scientific Computing (LNCC), Brazil.

Programme :
Programme détaillé :
Part I : Origin of the MHM Method (class 1)
Part II : MHM's Basic Theory (classes 2, 3 and 4)
Part III : Practical Aspects of the Implementation of the MHM Method (classes 4 and 5)

Pré-requis :
Basics of Finite Element methods and the theory of Partial Differential Equations.

Equipe pédagogique :
Frédéric Valentin (Senior Researcher LNCC, Petrópolis, Brésil), Antônio Tadeu Azevedo Gomes (Senior Researcher LNCC, Petrópolis, Brésil)

Compétences acquises à l'issue de la formation :
The participants will at the end of this formation have a broad and precise overview of the fundamental mechanism of this class of numerical methods. Furthermore, through practical sessions on computers, they will be able to put this theory into practice and run simulations on some examples.

La formation participe à l'objectif suivant :être directement utile pour la réalisation des travaux personnels de recherche