  We introduce new control-volume finite-element discretization schemes
suitable for solving the Stokes problem. Within a common framework, we present
different approaches for constructing such schemes. The first and most
established strategy employs a non-overlapping partitioning into control
volumes. The second represents a new idea by splitting into two sets of control
volumes, the first set yielding a partition of the domain and the second
containing the remaining overlapping control volumes required for stability.
The third represents a hybrid approach where finite volumes are combined with
finite elements based on a hierarchical splitting of the ansatz space. All
approaches are based on typical finite element function spaces but yield
locally mass and momentum conservative discretization schemes that can be
interpreted as finite volume schemes. We apply all strategies to the inf-sub
stable MINI finite-element pair. Various test cases, including convergence
tests and the numerical observation of the boundedness of the number of
preconditioned Krylov solver iterations, as well as more complex scenarios of
flow around obstacles or through a three-dimensional vessel bifurcation,
demonstrate the stability and robustness of the schemes.
