  The Stokes equation posed on surfaces is important in some physical models,
but its numerical solution poses several challenges not encountered in the
corresponding Euclidean setting. These include the fact that the velocity
vector should be tangent to the given surface and the possible presence of
degenerate modes (Killing fields) in the solution. We analyze a surface finite
element method which provides solutions to these challenges. We consider an
interior penalty method based on the well-known Brezzi-Douglas-Marini $H({\rm
div})$-conforming finite element space. The resulting spaces are tangential to
the surface, but require penalization of jumps across element interfaces in
order to weakly maintain $H^1$ conformity of the velocity field. In addition
our method exactly satisfies the incompressibility constraint in the surface
Stokes problem. Secondly, we give a method which robustly filters Killing
fields out of the solution. This problem is complicated by the fact that the
dimension of the space of Killing fields may change with small perturbations of
the surface. We first approximate the Killing fields via a Stokes eigenvalue
problem and then give a method which is asymptotically guaranteed to correctly
exclude them from the solution. The properties of our method are rigorously
established via an error analysis and illustrated via numerical experiments.

Numerical Analysis;Numerical Analysis