  This paper addresses the construction and analysis of a class of domain
decomposition methods for the iterative solution of the quasi-static Biot
problem in three-field formulation. The considered discrete model arises from
time discretization by the implicit Euler method and space discretization by a
family of strongly mass-conserving methods exploiting $H^{div}$-conforming
approximations of the solid displacement and fluid flux fields. For the
resulting saddle-point problem, we construct monolithic overlapping domain
decomposition (DD) methods whose analysis relies on a transformation into an
equivalent symmetric positive definite system and on stable decompositions of
the involved finite element spaces under proper problem-dependent norms.
Numerical results on two-dimensional test problems are in accordance with the
provided theoretical uniform convergence estimates for the two-level
multiplicative Schwarz method.
