Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method

TL;DR

This paper introduces a new framework using the smoothing cofactor method to calculate spline space dimensions over rectilinear partitions, extending existing theories and validating with numerical examples.

Contribution

It extends dimension theory for polynomial splines by introducing TE-connected components and a new class of partitions, enabling explicit dimension calculations.

Findings

Dimension of spline spaces can be computed via conformality matrices.

Schumaker's lower bound is attainable in certain partition configurations.

Numerical examples validate the framework's effectiveness for various partitions.

Abstract

This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions.