Supersmoothness of multivariate splines

Polynomial splines over simplicial meshes in R^n (triangulations in 2D, tetrahedral meshes in 3D, and so on) sometimes have extra orders of smoothness at a vertex. This property is known as supersmoothness, and plays a role both in the construction of macroelements and in the finite element method.
Supersmoothness depends both on the number of simplices that meet at the vertex and their geometric configuration.

In this talk we review what is known about supersmoothness of polynomial splines and then discuss the more general setting of splines whose individual pieces are any infinitely smooth functions.

This is joint work with Kaibo Hu.