Analysis-suitable spline spaces of arbitrary degree on unstructured quadrilateral meshes by

We build C1 spline spaces, Sp(M), on unstructured quadrilateral meshes M, where multiple extraordinary points per element are allowed. These spline spaces can be used to parameterize non-planar geometries of arbitrary topology, and are very useful for the purpose of performing isogeometric analysis on such geometries. Our methodology is simple and is presented for splines of arbitrary bi-degree (p, p). The search for an analysis-suitable construction leads us first to an interesting spline space of reduced smoothness, Sp(M). The elements of Sp(M) are C0 across spoke edges of extraordinary points and C1 otherwise. We build basis functions for Sp(M) that are locally linearly independent, and we show that Sp(M) possesses optimal approximation properties for the Lq norms, 1  q  1; these qualities make Sp(M) itself an object worthy of analysis. Using Sp(M), we present the construction of Sp(M) whose elements are C1 smooth everywhere, and we prove that the splines spanning Sp(M) are linearly independent. The locality of our construction gives us confidence that the approximation properties of Sp(M) carry over to Sp(M), and we verify this conjecture numerically by solving function approximation problems. Our numerical results suggest optimal approximation in L2 and L1 norms for both Sp(M) and Sp(M). Finally, we demonstrate the applicability of Sp(M) to higher order partial di↵erential equations, with all numerical results exhibiting optimal convergence rates in L2, H1 and H2 norms.