  One of the important aspects of IsoGeometric Analysis (IGA) is the strong
link between Computer Aided Design and analysis. Two of IGA'a major challenge
are the assembly of patches (Constructive Solid Geometry geometries made of
Boolean primitives) and local refinement. In the present work we apply the
Additive Schwarz Domain Decomposition on overlapping domains. To leverage the
power of IGA we consider non matching meshes to solve PDEs on CSG assemblies.
As a side effect also we deal with local refinement .
  As a first step we study a collection of simple two domains problems, using
the software package GeoPDEs, all of them converged rapidly, even on very
distorted domains. For local refinement we implement a Chimera type of zooming
together with ASDDM iterations, and here again we get very attractive results.
A last effect is that the additive methods are naturally parallelizable and
thus can extend to large 3D problems. We give some examples of computations on
multi-patch bodies.
  This new implementation of the DD methods brings many interesting problems
such as: the definition of boundary conditions on trimmed patches, choice of
blending operators on the intersections , optimization of the overlap size,
choice of preconditioning to guarantee convergence ( as we do not have a
maximum principle here.)
