Analysis of a family of time-continuous strongly conservative space-time finite element methods for the dynamic Biot model

TL;DR

This paper introduces a family of space-time finite element methods for the dynamic Biot model, effectively capturing wave propagation and fluid-solid interactions in saturated porous media with proven optimal error estimates.

Contribution

It develops a novel variational space-time finite element framework combining continuous-in-time Galerkin and inf-sup stable DG approximations for the dynamic Biot model.

Findings

Proves optimal error estimates in energy and L2 norms.

Demonstrates stability and accuracy of the proposed methods.

Handles wave propagation phenomena in poroelastic media.

Abstract

We consider the dynamic Biot model 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. The 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 that combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with inf-sup stable $H(\rm{div})$-conforming approximations of discontinuous Galerkin (DG) type in case of the displacement and a mixed approximation of the flux, its time derivative and the pressure field. We prove error estimates in a combined energy norm as well as $L^2$~error estimates in space for the individual fields for both maximum and $L^2$ norm in time which are optimal for the displacement and pressure approximations.