Finite Element de Rham and Stokes Complexes in Three Dimensions

TL;DR

This paper systematically constructs finite element de Rham and Stokes complexes in three dimensions, introducing new smooth scalar elements and conforming vector elements, ensuring stability and exactness for advanced numerical simulations.

Contribution

It introduces a systematic construction of finite element complexes with various smoothness in 3D, including new smooth scalar elements and conforming vector elements, with proven stability and exactness.

Findings

Constructed smooth scalar finite elements in 3D.

Developed H(div) and H(curl)-conforming finite elements.

Established div stability and complex exactness.

Abstract

Finite element de Rham complexes and finite element Stokes complexes with various smoothness in three dimensions are systematically constructed. First smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. Based on the smooth scalar finite elements, both H(div)-conforming finite elements and H(curl)-conforming finite elements with various smoothness are devised, which induce the finite element de Rham complexes with various smoothness and the associated commutative diagrams. The div stability is established for the H(div)-conforming finite elements, and the exactness of these finite element complexes.