A perturbed threefold saddle-point formulation yielding new mixed finite element methods for poroelasticity with reduced symmetry

Abstract

We propose and analyze new mixed finite element methods for the linear poroelasticity problem, which models the coupled phenomena of fluid diffusion and solid deformation. The formulation is based on the introduction of the vorticity and the strain tensor as auxiliary unknowns, which makes it possible to eliminate the fluid content from the system. The latter can then be recovered from the strain tensor and the pressure. Then, by incorporating a multiple of the pressure gradient as an additional unknown, we arrive at an operator equation showing a threefold saddle-point structure, which, in turn, is perturbed by a term depending on the pressure variable. The well-posedness of the continuous formulation is established through a suitable extension of the usual Babu\v{s}ka--Brezzi theory, which yields a new abstract result, along with a recently developed approach to analyze perturbed saddle-point problems. The discrete analysis follows a similar strategy, employing arbitrary finite element spaces satisfying suitable assumptions. In particular, we provide concrete examples based on PEERS elements and derive the corresponding convergence rates. Finally, several numerical experiments are presented, which confirm the theoretical results and illustrate the good performance of the methods.