Quasi-interpolation projectors for subdivision function spaces

Subdivision surfaces as an extension of splines have become a promising technique for addressing PDEs on models with complex topologies in isogeometric analysis. This has sparked interest in exploring the approximation by subdivision function spaces. Quasi-interpolation serves as a significant tool in the field of approximation, offering benefits such as low computational expense and strong numerical stability. In this paper, we propose a straightforward approach for constructing the quasi-interpolation projectors of subdivision function spaces that features explicit formulations and achieves a highly desirable approximation order. The local interpolation problem is constructed based on the subdivision mask and the limit position mask, overcoming the cumbersome evaluation of the subdivision basis functions and the difficulty associated with deriving explicit solutions to the problem. Explicit quasi-interpolation formulas for the loop, modified loop, and Catmull–Clark subdivisions are provided. Numerical experiments demonstrate that these quasi-interpolation projectors achieve an expected approximate order and present promising prospects in isogeometric collocation.