  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 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. Our arguments can however be extended to three- and four field
formulations and other time discretization strategies.
