De Rham Complexes for Weak Galerkin Finite Element Spaces

TL;DR

This paper introduces two de Rham complex sequences for weak Galerkin finite element spaces on polyhedral meshes, demonstrating their commutativity and establishing exactness for the lowest order case.

Contribution

It develops two new de Rham complex sequences for weak Galerkin finite element methods on polyhedral elements, with proofs of their properties and exactness.

Findings

Both complexes commute on general polyhedral elements.

Exactness is proven for the lowest order complex.

The complexes accommodate polynomials of equal and descending orders.

Abstract

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element methods on general polyhedral elements. One of the sequences uses polynomials of equal order for all the finite element spaces involved in the sequence and the other one uses polynomials of naturally decending orders. It is shown that the diagrams in both de Rham complexes commute for general polyhedral elements. The exactness of one of the complexes is established for the lowest order element.