Discrete Elasticity Exact Sequences on Worsey-Farin Splits

TL;DR

This paper develops a new finite element elasticity complex on Worsey-Farin splits in 3D, featuring the first symmetric stress element with minimal degrees of freedom and no extrinsic supersmoothness.

Contribution

It introduces the first strongly symmetric stress finite element in three dimensions with no vertex or edge degrees of freedom and minimal polynomial degree.

Findings

Constructed conforming finite element elasticity complexes on Worsey-Farin splits.

Developed unisolvent degrees of freedom and commuting projections.

Achieved the lowest polynomial degree for stress spaces with no extrinsic supersmoothness.

Abstract

We construct conforming finite element elasticity complexes on Worsey-Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey-Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.