Seminário de Avaliação - Série A: Hybrid and Discontinuous FEM 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:
Abimael Fernando Dourado Loula - Laboratório Nacional de Computação Científica - LNCC (presidente)
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Eduardo Gomes Dutra do Carmo - Universidade Federal do Rio de Janeiro - UFRJ
Maicon Ribeiro Correa - UNICAMP

Suplentes:
Regina Célia Cerqueira de Almeida - Laboratório Nacional de Computação Científica - LNCC

Resumo:

New Hybrid and Discontinuous FEM methods for porous media flow problems are proposed. Two classes of metho ds 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 formulations. For mixed methods, a similar procedure is applied to obtain new formulations considering Least-Squares residuals. The hybridization follows the well-known HDG methods imposing local int erface 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 method, which uses two Lagrange multipliers, 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 l2-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. To demonstrate the applicability of the new formulations, several porous media flow problems are solved: (i) Darcy’s flow in heterogeneous porous media, (ii) transport problem triggering instabilities, and (iii) double porosity/permeability Darcy’s flow problems.