We consider the systematic numerical approximation of Biot’s quasistatic model for the consolidation of a poroelastic medium. Various discretization schemes have been analysed for this problem and inf-sup stable finite elements have been found suitable to avoid spurios pressure oscillations in the initial phase of the evolution. In this paper, we first clarify the role of the inf-sup condition for the well-posedness of the continuous problem and discuss the choice of appropriate initial conditions. We then develop an abstract error analysis that allows us to analyse some approximation schemes discussed in the literature in a unified manner. In addition, we propose and analyse the high-order time discretization by a scheme that can be interpreted as a variant of continuous-Galerkin or particular Runge–Kutta methods applied to a modified system. The scheme is designed to preserve both, the underlying differential–algebraic structure and the energy-dissipation property of the problem. In summary, we obtain high-order Galerkin approximations with respect to space and time and derive order-optimal convergence rates. The numerical analysis is carried out in detail for the discretization of the two-field formulation by Taylor-Hood elements and a variant of a Runge–Kutta time discretization.

我们考虑了Biot准静态模型对于多孔弹性介质固结的系统数值近似。已针对该问题分析了各种离散化方案，并发现了ins-up稳定有限元可避免在演化的初始阶段出现尖峰压力振荡。在本文中，我们首先阐明了连续条件对连续问题的适定性的作用，并讨论了适当初始条件的选择。然后，我们开发一种抽象的误差分析，使我们能够以统一的方式分析文献中讨论的一些近似方案。此外，我们提出并分析了一种高阶时间离散化方案，该方案可以解释为应用于改进系统的连续Galerkin方法或特定Runge-Kutta方法的变体。该方案旨在保留底层的微分代数结构和问题的耗能特性。总而言之，我们获得关于空间和时间的高阶Galerkin近似，并得出阶次最优收敛速度。通过Taylor-Hood元素和Runge-Kutta时间离散化的变体，对两场公式的离散化进行了详细的数值分析。