A family of finite element Stokes complexes in three dimensions

TL;DR

This paper develops new finite element complexes in 3D that enable stable and convergent algorithms for Stokes and gradcurl problems, with practical numerical validation.

Contribution

It introduces a family of finite element Stokes complexes on tetrahedral meshes, including gradcurl-conforming elements and stable Stokes pairs, advancing finite element methods in 3D.

Findings

Finite element Stokes complexes constructed in 3D.

Stable algorithms for Stokes and gradcurl problems demonstrated.

Numerical experiments confirm convergence and stability.

Abstract

We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition. We demonstrate the validity of the algorithms by numerical experiments.