  Recently a nonconforming surface finite element was developed to discretize
2D vector-valued compressible flow problems in a 3D domain. In this
contribution we derive an error analysis for this approach on a vector-valued
Laplace problem, which is an important operator for fluid-equations on the
surface. In our setup, the problem is approximated via edge-integration on
local flat triangles using the nonconforming linear Crouzeix-Raviart element.
The flat planes coincide with the surface at the edge midpoints. This is also
the place, where the Crouzeix-Raviart element requires continuity between two
neighbouring elements. The developed Crouzeix-Raviart approximation is a
non-parametric approach that works on local coordinate systems, established in
each triangle. This setup is numerically efficient and straightforward to
implement. For this Crouzeix-Raviart discretization we derive optimal error
bounds in the $H^1$-norm and $L^2$-norm and present an estimate for the
geometric error. Numerical experiments validate the theoretical results.
