High order space and time discretizations to biot’s consolidation problem

• Ano de defesa: 2022
• Autor(a) principal: Ledoino, Ismael de Souza
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Laboratório Nacional de Computação Científica; Coordenação de Pós-Graduação e Aperfeiçoamento (COPGA); Brasil; LNCC; Programa de Pós-Graduação em Modelagem Computacional
• Palavras-chave em Português: Elasticidade; Materiais porosos; Galerkin, Métodos de; CNPQ::ENGENHARIAS
• Link de acesso: https://tede.lncc.br/handle/tede/309

We propose high order space and time discretizations 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 based on a mixed formulation in three main unknowns: displacement, pore pressure and hydrostatic pressure, introduced to generate locking-free finite element approximations. The hybridization is made through the insertion of continuous or discontinuous Lagrange multipliers, identified with the trace of the displacement and pore pressure fields at the skeleton of the mesh. The numerical analysis in space handles discontinuous Lagrange multipliers of reduced order, which are associated to projection operators to preserve optimal rates of convergence. 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 derived GLC method leads to a more efficient way of implementing Runge-Kutta schemes for PDEs, and the conversion formula is applicable to many types of RK methods. This includes the Radau IIA collocation method, used to maintain the locking-free and oscillatory-free behavior in high order approximations. A numerical analysis of the semi-discrete problem is presented, and error estimates are also derived for the fully discrete problem associated with implicit Euler and Crank-Nicolson schemes. Numerical experiments are performed to show the optimal convergence rates in space and time, the locking-free and oscillatory-free properties of the space discretization and the unconditional stability of the time discretization.