Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, pressure with non-constant viscosity homogeneous Dirichlet boundary condition velocity. The weak formulation includes least-squares arising from constitutive equation incompressibility condition, we show that it satisfies hypotheses Babu\vska-Brezzi theory. Repeating arguments continuous analysis, stability solvability discrete problem are established. is suited any Stokes inf-sup stable pair velocity pressure, while vorticity generic space (of arbitrary order) can be used. A priori a posteriori error estimates derived using two specific families subspaces. Finally, provide set numerical tests illustrating behaviour scheme, verifying theoretical convergence rates, showing performance adaptive algorithm guided by residual estimation.