  We consider a surface Stokes problem in stream function formulation on a
simply connected oriented surface $\Gamma \subset \mathbb{R}^3$ without
boundary. This formulation leads to a coupled system of two second order scalar
surface partial differential equations (for the stream function and an
auxiliary variable). To this coupled system a trace finite element
discretization method is applied. The main topic of the paper is an error
analysis of this discretization method, resulting in optimal order
discretization error bounds. The analysis applies to the surface finite element
method of Dziuk-Elliott, too. We also investigate methods for reconstructing
velocity and pressure from the stream function approximation. Results of
numerical experiments are included.
