Defesa de Tese de Doutorado: Hybrid and Discontinuous Finite Element Methods for porous media flow problems

Orientadores:
Abimael Fernando Dourado Loula - Laboratório Nacional de Computação Científica - LNCC
Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC (presidente)
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Regina Célia Cerqueira de Almeida - Laboratório Nacional de Computação Científica - LNCC
Maicon Ribeiro Correa - UNICAMP
Eduardo Gomes Dutra do Carmo - Universidade Federal do Rio de Janeiro - UFRJ

Suplentes:
Sandra Mara Cardoso Malta - Laboratório Nacional de Computação Científica - LNCC
Álvaro Luiz Gayoso de A zevedo Coutinho - Universidade Federal do Rio de Janeiro - COPPE/UFRJ

Resumo:

New Hybrid and Discontinuous FEM formulations for porous media flow problems are proposed. Two classes of methods are devised: approximations (i) based on Least-Squares variational principles and (ii) combining classical Galerkin forms with Least-Squares residuals. Both primal and mixed discontinuous approximations are obtained following the same central ideas of the two approaches. The formulations are then hybridized, resulting in new hybrid discontinuous methods. Through this technique, low-order primal least-squares methods that were reported as impractical -- due to the regularity requirements and the associated computational cost as a consequence of the polynomial degrees in the element interior -- are now amended using static condensation. Primal Least-Squares formulations are related to classical primal hybrid methods, through symmetrization of the former, giving rise to adjoint-consistent f ormulations. A similar procedure is applied for mixed methods to obtain new formulations considering Least-Squares residuals. The hybridization follows the well-known HDG methods imposing local interface conditions (transmission conditions on the mesh skeleton). In all methods, the Lagrange multiplier is identified as the trace of the primal variable, resulting in a reduced number of unknowns when compared with the Least-Squares Weak Galerkin or Discontinuous Galerkin methods, for instance. Considering a general scalar elliptic problem, we performed numerical experiments using the method of manufactured solutions to show that the new formulations have optimal convergence rates -- in terms of $L^2$-norm -- for primal and flux (for mixed methods) variables. Additionally, the conditioning of resulting algebraic systems is assessed, showing similar results as in HDG even for nonsymmetric systems. The new formulations are applied to solve several porous media flow problems: (i) Darcy's f low in heterogeneous porous media, (ii) transport problem triggering instabilities, and (iii) double porosity/permeability Darcy's flow problems.