Spline Spaces over Quadrangle Meshes with Complex Topologies

We present a new type of spline functions defined over a quadrangular mesh equipped with an equivalence relation, in such a way that physical spaces with a complex topology can be represented as an homomorphic image of such meshes. We provide general definitions, a dimension formula for a subclass of these spline spaces, an explicit construction of their bases and also a process for local refinement. These developments, motivated by plane curvilinear mesh constructions are illustrated on several parametrization problems. Our main target in these constructions is to approximate isobaric lines of magnetic fields encountered in MHD simulation for Tokamaks. Their particularity is that one of the isobaric curve has a node singularity.