Exame de Qualificação: 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:
Pablo Javier Blanco - Laboratório Nacional de Computação Científica - LNCC (presidente)
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Abimael Fernando Dourado Loula - 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 OPERATORS. THE ONE-LEVEL MH M 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.