Time-continuous strongly conservative space-time finite element methods for the dynamic Biot model

Abstract

We consider the dynamic Biot model (see [Biot, M. A. J. Appl. Phys. 33, 1482--1498 (1962)]) describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. This model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest. A family of variational space-time finite element methods is proposed, which combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with $H(\mathrm{div})$-conforming approximations of the displacement field, its time derivative, and the flux field--of discontinuous Galerkin (DG) type for displacements--with a piecewise polynomial pressure approximation, providing an inf-sup stable strongly conservative mixed method in each case. We prove error estimates in a combined energy norm in space for the maximum norm in time. The theoretical results are confirmed by numerical experiments for different polynomial orders in space and time.