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

Bingru Huang

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arXiv · 2508.14649

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

Bingru Huang

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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.

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Bingru Huang. 2025-08-20. Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective. https://arxiv.org/abs/2508.14649

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