Exact sequences on Powell-Sabin splits
For researchers in finite element methods, this work provides a theoretical framework for constructing exact sequences on Powell-Sabin splits, enabling stable discretizations of the Stokes problem.
The paper constructs smooth finite element spaces on Powell-Sabin triangulations that form an exact sequence, with the first space being the classical C1 Powell-Sabin space. The resulting spaces provide stable and divergence-free pairs for the Stokes problem, with commuting projections.
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.