  Compatible Discrete Operator schemes preserve basic properties of the
continuous model at the discrete level. They combine discrete differential
operators that discretize exactly topological laws and discrete Hodge operators
that approximate constitutive relations. We devise and analyze two families of
such schemes for the Stokes equations in curl formulation, with the pressure
degrees of freedom located at either mesh vertices or cells. The schemes ensure
local mass and momentum conservation. We prove discrete stability by
establishing novel discrete Poincar\'e inequalities. Using commutators related
to the consistency error, we derive error estimates with first-order
convergence rates for smooth solutions. We analyze two strategies for
discretizing the external load, so as to deliver tight error estimates when the
external load has a large irrotational or divergence-free part. Finally,
numerical results are presented on three-dimensional polyhedral meshes.
