  This paper is concerned with using discontinuous Galerkin isogeometric
analysis (dGIGA) as a numerical treatment of Diffusion problems on orientable
surfaces $\Omega \subset \mathbb{R}^3$. The computational domain or surface
considered consist of several non-overlapping sub-domains or patches which are
coupled via an interior penalty scheme. In Langer and Moore U. Langer and S. E.
Moore,2014, we presented a priori error estimate for conforming computational
domains with matching meshes across patch interface and a constant diffusion
coefficient. However, in this article, we generalize the \textit{a priori}
error estimate to non-matching meshes and discontinuous diffusion coefficients
across patch interfaces commonly occurring in industry. We construct B-Spline
or NURBS approximation spaces which are discontinuous across patch interfaces.
We present \textit{a priori} error estimate for the symmetric discontinuous
Galerkin scheme and numerical experiments to confirm the theory.
