  We develop a cut finite element method for the Darcy problem on surfaces. The
cut finite element method is based on embedding the surface in a three
dimensional finite element mesh and using finite element spaces defined on the
three dimensional mesh as trial and test functions. Since we consider a partial
differential equation on a surface, the resulting discrete weak problem might
be severely ill conditioned. We propose a full gradient and a normal gradient
based stabilization computed on the background mesh to render the proposed
formulation stable and well conditioned irrespective of the surface positioning
within the mesh. Our formulation extends and simplifies the Masud-Hughes
stabilized primal mixed formulation of the Darcy surface problem proposed in
[28] on fitted triangulated surfaces. The tangential condition on the velocity
and the pressure gradient is enforced only weakly, avoiding the need for any
tangential projection. The presented numerical analysis accounts for different
polynomial orders for the velocity, pressure, and geometry approximation which
are corroborated by numerical experiments. In particular, we demonstrate both
theoretically and through numerical results that the normal gradient stabilized
variant results in a high order scheme.
