  We develop non-overlapping domain decomposition methods for the Biot system
of poroelasticity in a mixed form. The solid deformation is modeled with a
mixed three-field formulation with weak stress symmetry. The fluid flow is
modeled with a mixed Darcy formulation. We introduce displacement and pressure
Lagrange multipliers on the subdomain interfaces to impose weakly continuity of
normal stress and normal velocity, respectively. The global problem is reduced
to an interface problem for the Lagrange multipliers, which is solved by a
Krylov space iterative method. We study both monolithic and split methods. In
the monolithic method, a coupled displacement-pressure interface problem is
solved, with each iteration requiring the solution of local Biot problems. We
show that the resulting interface operator is positive definite and analyze the
convergence of the iteration. We further study drained split and fixed stress
Biot splittings, in which case we solve separate interface problems requiring
elasticity and Darcy solves. We analyze the stability of the split
formulations. Numerical experiments are presented to illustrate the convergence
of the domain decomposition methods and compare their accuracy and efficiency.
