Study of Primal Hybrid Finite Element Solutions for the Biot System

Resumo

Poro-elasticity problems combine the deformation of a porous matrix with the flow of a fluid through its porous. The most classical linear model for such a problem is the Biot system of partial differential equations, where the Darcy law is used to model the flow, linear elasticity is assumed to the poromechanics, and the Terzaghi principle is assumed to be valid. In the context of Finite Element approximations for Biot, H1-conforming and mixed strategies are widely discussed in the literature. Hybrid methods, on the other hand, have not received the same level of attention. In this work, we discuss a Primal hybrid method for the Biot system, where Lagrange multipliers are added to weakly enforce the continuity of the pressure and displacement fields. The main advantage of the proposed strategy, in comparison with H1-conforming methods, is the better mass/equilibrium conservation properties of the discrete solutions, which allows for an element-wise post-processing strategy to recover the stress and the velocity fields, leading to global H(div)-conforming approximations. The recovered stress can be either the effective or the total stress, depending on the choices made in the hybridization process. In this work, we present stable approximation spaces for the proposed method in quadrilateral meshes, allowing different approximation orders for the elasticity and the flow subproblems. We also discuss the factors that influence the H(div) convergence for the recovered velocity and stress.