# Properties of Spline Spaces Over Structured Hierarchical Box Partitions

## TL;DR

This paper explores how local refinements in hierarchical spline spaces can optimize the number of basis functions and improve numerical stability in computational applications.

## Contribution

It demonstrates that local one-directional refinements in LRB spline spaces can reduce basis functions and enhance matrix condition numbers.

## Key findings

- Refinements can minimize B-splines per element to polynomial degree limits.
- Local modifications improve the condition numbers of mass and stiffness matrices.
- LRB spaces can be constructed to contain and extend THB spline spaces.

## Abstract

Given a spline space spanned by Truncated Hierarchical B-splines (THB), it is always possible to construct a spline space spanned by Locally Refined B-splines (LRB) that contains the THB-space. Starting from configurations where the two spline spaces are equal, we adress what happens to the properties of the LRB-space when it is modified by local one-directional refinement at convex corners of, and along edges between dyadic refinement regions. We show that such local modifications can reduce the number of B-splines over each element to the minimum prescribed by the polynomial bi-degree, and that such local refinements can be used for improving the condition numbers of mass and stiffness matrices.