H1-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes

TL;DR

This paper introduces a finite element cochain complex on Cartesian meshes that ensures H1-conformity and commuting projectors, generalizing to arbitrary order, enhancing stability and compatibility in finite element methods.

Contribution

It presents a novel H1-conforming finite element cochain complex on Cartesian meshes with commuting projectors, generalized to any order, improving finite element analysis tools.

Findings

Constructed H1-conforming finite element spaces with exterior derivatives in H1

Developed L2-stable projectors that commute with exterior derivatives

Generalized the finite element complex to arbitrary order

Abstract

A finite element cochain complex on Cartesian meshes of any dimension based on the H1-inner product is introduced. It yields H1-conforming finite element spaces with exterior derivatives in H1. We use a tensor product construction to obtain L2-stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.