A Study of Two Modes of Locking in Poroelasticity

In this paper, we study two modes of locking phenomena in poroelasticity: Possion locking and pressure oscillations. We first study the regularity of the solution of the Biot model to gain some insight into the cause of Poisson locking and show that the displacement gets into a divergence-free state as the Lamé constant $\lambda \to \infty$. We also examine the cause of pressure oscillations from an algebraic point of view when a three-field mixed finite element method is used. Based on the results of our study on the causes of the two modes of locking, we propose a new family of mixed finite elements that are free of both pressure oscillations and Poisson locking. Some numerical results are presented to validate our theoretical studies.