Hierarchical B-splines on regular triangular partitions

Multivariate splines have a wide range of applications in function approximation, finite element analysis and geometric modeling. They have been extensively studied in the last several decades, and specially the theory on bivariate B-splines over regular triangular partition is well developed. However, the above mentioned splines do not have local refinement property – a property that is very important in adaptive function approximation and level of detailed representation of geometric models. In this paper, we introduce the concept of hierarchial bivariate splines over regular triangular partitions and construct basis functions of such spline space that satisfy some nice properties. We provide some examples of hierarchical splines over triangular partitions in surface fitting and in solving numerical PDEs, and the results turn out to be promising.