Defesa de Dissertação de Mestrado: High Order Space and Time Discretizations to Biot’s Consolidation Problem

Orientadores:
Abimael Fernando Dourado Loula - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Abimael Fernando Dourado Loula - Laboratório Nacional de Computação Científica - LNCC (presidente)
Sandra Mara Cardoso Malta - Laboratório Nacional de Computação Científica - LNCC
Maicon Ribeiro Correa - UNICAMP
Cristiane Oliveira de Faria - Laboartório Nacional de Computação Científica - LNCC
Eduardo Gomes Dutra do Carmo - Universidade Federal do Rio de Janeiro - UFRJ

Suplentes:
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Rodrigo Weber dos Santos - Universidade Federal de Juiz de Fora - UFJF

Resumo:

We propose a high order space and time discretization to Biot’s consolidation problem, composed of a mixed hybrid discontinuous Galerkin (HDG) method in space and a modified Runge-Kutta (MRK) method in time. The HDG method is a mixed problem in three main unknowns: displacement, pore pressure and hydrostatic pressure, being the latter introduced mainly for the purpose of generating a locking-free method. The hybridization is made through the insertion of continuous (EDG) or discontinuous (HDG) Lagrange multipliers identified with the trace of the displacement and pore pressure fields at the skeleton of the mesh. Our analysis in space handles reduced order discontinuous Lagrange multipliers, which are possible with an use of projection operators. For the discretization in time, we generalize the second order convergent Crank-Nicolson scheme to an arbitrary high order convergent Runge-Kutta scheme, known as Gauss-Legendre collocation method (GLC). The way we derive the GLC method suggests a more efficient way of implementing it for PDEs, and the conversion formula is applicable to many other RK methods, including some explicit ones. The numerical analysis of the semi-discrete problem is devised, and numerical estimates are also obtained for the fully discrete problem in which implicit Euler and Crank-Nicolson schemes are used.
Numerical experiments are performed to show the optimal convergence rates in space and time, the locking-free property of the space discretization and the unconditional stability of the time discretization.