New degrees of freedom for differential forms on cubical meshes

TL;DR

This paper introduces new degrees of freedom for higher order differential forms on cubical meshes, expanding the discrete exterior calculus framework and enabling higher order methods on cubical grids.

Contribution

It develops a novel approach to define unisolvent degrees of freedom for higher order differential forms on cubical meshes using small cubes, inspired by Whitney forms.

Findings

Small cubes provide unisolvent degrees of freedom.

The approach is compatible with discrete exterior calculus.

It extends higher order methods to cubical meshes.

Abstract

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Significantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.