  In this paper, we propose a numerical method for the solution of
time-dependent flow problems in mixed form. Such problems can be efficiently
approximated on hierarchical grids, obtained from an unstructured coarse
triangulation by using a regular refinement process inside each of the initial
coarse elements. If these elements are considered as subdomains, we can
formulate a non-overlapping domain decomposition method based on the
lowest-order Raviart-Thomas elements, properly enhanced with Lagrange
multipliers on the boundaries of each subdomain (excluding the Dirichlet
edges). A suitable choice of mixed finite element spaces and quadrature rules
yields a cell-centered scheme for the pressures with a local 10-point stencil.
The resulting system of differential-algebraic equations is integrated in time
by the Crank-Nicolson method, which is known to be a stiffly accurate scheme.
As a result, we obtain independent subdomain linear systems that can be solved
in parallel. The behaviour of the algorithm is illustrated on a variety of
numerical experiments.
