Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures (Communications in Computational Physics, 2018)
Motivated by the magneto hydrodynamic (MHD) simulation for Tokamakswith Isogeometric analysis, we present splines defined over a rectangular mesh with acomplex topological structure, i.e., with extraordinary vertices. These splines are piecewisepolynomial functions of bi-degree (d,d) and $C^r$ parameter continuity. And wecompute their dimension and exhibit basis functions called Hermite bases for bicubicspline spaces. We investigate their potential applications for solving partial differentialequations (PDEs) over a physical domain in the framework of Isogeometric analysis.For instance, we analyze the property of approximation of these spline spaces for the $L^2$-norm; we show that the optimal approximation order and numerical convergencerates are reached by setting a proper parameterization, although the fact that the basisfunctions are singular at extraordinary vertices.