# Exact sequences on Powell-Sabin splits

## TL;DR

This paper develops exact sequence finite element spaces on Powell-Sabin triangulations, including $C^1$ spaces and divergence-free pairs, with commuting projections for stable Stokes problem discretization.

## Contribution

It introduces new finite element spaces forming an exact sequence on Powell-Sabin splits, with compatible degrees of freedom and projections for fluid dynamics applications.

## Key findings

- Constructed smooth finite element spaces forming an exact sequence
- Developed degrees of freedom inducing commuting projections
- Provided stable divergence-free pairs for the Stokes problem

## Abstract

We construct smooth finite elements spaces on Powell-Sabin triangulations that form an exact sequence. The first space of the sequence coincides with the classical $C^1$ Powell-Sabin space, while the others form stable and divergence-free yielding pairs for the Stokes problem. We develop degrees of freedom for these spaces that induce projections that commute with the differential operators.