  Isogeometric analysis uses the same class of basis functions for both,
representing the geometry of the computational domain and approximating the
solution. In practical applications, geometrical patches are used in order to
get flexibility in the geometrical representation. This patch representation
corresponds to a domain decomposition. In this paper, we present for the first
time a Discontinuous Galerkin (DG) Method that allows for discontinuities only
along the sub-domain (patch) boundaries. The required smoothness is obtained by
the DG terms associated with the boundary of the sub-domains. The construction
and corresponding discretization error analysis of such DG scheme is presented
for Elliptic PDEs in a 2D as well as on open and closed surfaces. Furthermore,
we present numerical results to confirm the theory presented.
