Low-order divergence-free approximations for the Stokes problem on Worsey-Farin and Powell-Sabin splits

TL;DR

This paper introduces low-order, divergence-free finite element methods for the Stokes problem using specialized mesh splits, ensuring stability and ease of implementation in 2D and 3D.

Contribution

It develops novel divergence-free finite element approximations on Worsey-Farin and Powell-Sabin splits, with simple velocity spaces and pressure spaces that handle weak continuity.

Findings

The methods are inf-sup stable and divergence-free.

Pressure constraints can be enforced algebraically.

Applicable to 2D and 3D mesh splits.

Abstract

We derive low-order, inf-sup stable and divergence-free finite element approximations for the Stokes problem using Worsey-Farin splits in three dimensions and Powell-Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, where as the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.