Seminário de Avaliação - Série A: Development and Analysis of Pre- and Post-Processing Techniques for Multiscale Methods Applied to Fluid Models

Orientadores:
Frédéric Gerard Christian Valentin - Laboratório Nacional de Computação Científica - LNCC
Weslley da Silva Pereira - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Frédéric Gerard Christian Valentin - Laboratório Nacional de Computação Científica - LNCC (presidente)
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC
Henrique de Melo Versieux - Universidade Federal de Minas Gerais - UFMG

Resumo:

The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method originally proposed for linear elliptic opera tors. The one-level MHM method is conforming in the H(div) space. However, when the multiscale basis functions are discretized using the standard Galerkin method on piecewise continuous polynomial space, it results in a mixed global problem in which the post-processed discrete dual variable no longer belongs to the H(div) space. Furthermore, the original MHM paper establishes a priori error estimates assuming local H2 regularity in the first-level mesh and an a posteriori error estimator without clean constants. So, in this work, we intend to contribute to improving (possibly some of) these points, and also formally extend the MHM to a non-linear problem. Notably, in this thesis, we propose the following:
i) A new flux reconstruction in the H(div) space for the MHM method applied to an elliptic equation;
ii) An a posteriori error estimator based on the flux equilibrated technique as a by-product of item (i);
iii) An elliptic version of the MHM method, which leads to a system of positive definite matrices and from which convergence may be proved under conditions of moderate regularity of the exact solution on the physical partition;
iv) A version of the MHM method for nonlinear Diffusive Wave (DW) equation, a model commonly used in the modeling of flooding.