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.

翻译：本文考虑描述饱和多孔介质中流体流动与固体变形相互作用的动态Biot模型，该模型包含液相和固相中的波动传播现象。该模型将动量守恒的双曲型方程、时间二阶动态达西定律与质量守恒的抛物型方程相结合，本文采用三场公式化表述，以弹性矩阵位移、流体速度和流体压力作为关注物理场。我们提出了一族变分时空有限元方法，该方法将任意多项式阶次的连续时间Galerkin格式与位移场的inf-sup稳定$H(\rm{div})$相容间断Galerkin（DG）型近似，以及通量场、其时间导数和压力场的混合近似相结合。我们证明了组合能量范数下的误差估计，以及对于各场在空间上的$L^2$误差估计（时间上分别采用最大范数和$L^2$范数），这些估计对于位移和压力逼近而言是最优的。