Spline spaces over rectangular meshes with arbitrary topologies and its application to the Grad-Shafranov equation

Motivated by the magneto hydrodynamic (MHD) simulation for tokamaks with an isoparametric finite element method or isogeometric analysis, we present a new type of spline space defined over a rectangular mesh with arbitrary topology. A set of bases called Hermite bases is constructed and applied to solving the Grad-Shafranov equation which is the equilibrium in the resistive MHD model. H 1 integrability assumption is used for designing parameterizations of the examples. Because the Grad-Shafranov equation is the second order PDE and there are isolated singularities of the parameterizations generally. To validate the continuity of the numerical solution of the Grad-Shafranov equation and its gradients on the physical domain, the errors between the exact solution and the numerical solution are compared with the L 2-norm and H 1-norm. The optimal convergence rates are reached.