Dimensions and bases of hierarchical tensor-product splines

Abstract

We prove that the dimension of trivariate tensor-product spline space of tri-degree (m, m, m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Juttler for two-dimensional hierarchical meshes. (C) 2013 Elsevier B.V. All rights reserved.