  We consider a Stokes problem posed on a 2D surface embedded in a 3D domain.
The equations describe an equilibrium, area-preserving tangential flow of a
viscous surface fluid and serve as a model problem in the dynamics of material
interfaces. In this paper, we develop and analyze a Trace finite element method
(TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element
spaces defined on a fixed, surface-independent background mesh which consists
of shape-regular tetrahedra. Thus, there is no need for surface parametrization
or surface fitting with the mesh. The TraceFEM treated here is based on $P_1$
bulk finite elements for both the velocity and the pressure. In order to
enforce the velocity vector field to be tangential to the surface we introduce
a penalty term. The method is straightforward to implement and has an $O(h^2)$
geometric consistency error, which is of the same order as the approximation
error due to the $P_1$--$P_1$ pair for velocity and pressure. We prove
stability and optimal order discretization error bounds in the surface $H^1$
and $L^2$ norms. A series of numerical experiments is presented to illustrate
certain features of the proposed TraceFEM.
