Inf-sup stable finite elements on barycentric refinements producing divergence--free approximations in arbitrary dimensions

TL;DR

This paper introduces new finite element pairs on barycentric refinements that are stable and produce divergence-free velocity approximations for the Stokes problem in any dimension, applicable to arbitrary polynomial degrees.

Contribution

The authors develop inf-sup stable finite element pairs that ensure divergence-free solutions in arbitrary dimensions and polynomial degrees, a significant advancement in finite element methods for incompressible flows.

Findings

Stable finite element pairs constructed for arbitrary dimensions.

Divergence maps velocity space onto pressure space, ensuring divergence-free solutions.

Local inf-sup stability proven for all polynomial degrees and dimensions.

Abstract

We construct several stable finite element pairs for the Stokes problem on barycentric refinements in arbitrary dimensions. A key feature of the spaces is that the divergence maps the discrete velocity space onto the the discrete pressure space; thus, when applied to models of incompressible flows, the pairs yield divergence-free velocity approximations. The key result is a local inf-sup stability that holds for any dimension and for any polynomial degree. With this result, we construct global divergence-free and stable pairs in arbitrary dimension and for any polynomial degree.