Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective

TL;DR

This paper presents a new semi-explicit method for constructing $C^r$ basis functions for polynomial splines over arbitrary planar partitions, including the first basis for Morgan-Scott partitions, addressing dimensional instability issues.

Contribution

It introduces a semi-explicit basis construction method that overcomes previous limitations and constructs bases on dimensionally unstable meshes, including the Morgan-Scott partition.

Findings

Successfully constructed basis functions for Morgan-Scott partition

Resolved theoretical challenges in spline space basis construction

Provided a comprehensive comparison of basis construction methods

Abstract

This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on dimensionally unstable meshes. We provide a comprehensive comparison of basis construction methods, classifying them as explicit, semi-explicit, and implicit. Our method, a semi-explicit construction using Extended Edge Elimination conditions, uniquely resolves all theoretical challenges in spline spaces by ensuring a complete basis. For the first time, we construct basis functions for the spline space over the Morgan-Scott partition, previously unachieved, and elucidate dimensional instability through this construction.