Seminário de Avaliação - Série A: High Order RK-HDG Discretizations for 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)
Alexandre Loureiro Madureira - Laboratório Nacional de Computação Científica - LNCC
Sandra Mara Cardoso Malta - Laboratório Nacional de Computação Científica - LNCC

Suplentes:
Eduardo Gomes Dutra do Carmo - Universidade Federal do Rio de Janeiro - UFRJ
Maicon Ribeiro Correa - UNICAMP

Resumo:

We propose a space-time discretization for Biot’s consolidation problem which is composed of a mixed hybrid discontinuous Galerkin method in space and a Runge-Kutta method in time. The discretization is arbitrarily high order in space and time, and the time discretization scheme is applicable to any other time-dependent problem. The discretization in space consists of 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 (GL). The way we derive the GL method suggests a more efficient way of implementing it for PDEs, and t he conversion formula is applicable to many other Runge-Kutta 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.