Finite element de Rham and Stokes complexes in three dimensions

Abstract

Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H ( div ) H(\operatorname {div}) -conforming finite elements and H ( curl ) H(\operatorname {curl}) -conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H ( div ) H(\operatorname {div}) -conforming finite elements is established, and the exactness of these finite element complexes is proven.