Cubic Lagrange elements satisfying exact incompressibility

TL;DR

This paper demonstrates the stability of cubic Lagrange finite elements satisfying incompressibility conditions on specific meshes, providing insights into divergence characterization and its relation to higher-order polynomial spaces.

Contribution

It introduces an inf-sup stable cubic Lagrange element for incompressible flows and characterizes divergence and space dimensions on specialized meshes.

Findings

Inf-sup stability of cubic elements on certain meshes

Characterization of divergence in velocity space

Relation to C^1 quartic space dimensions

Abstract

We prove that an analog of the Scott-Vogelius finite elements are inf-sup stable on certain nondegenerate meshes for piecewise cubic velocity fields. We also characterize the divergence of the velocity space on such meshes. In addition, we show how such a characterization relates to the dimension of C^1 piecewise quartics on the same mesh.