The Bubble Transform and the de Rham Complex

TL;DR

This paper extends the bubble transform technique to differential forms, enabling decomposition into local bubbles that commute with exterior derivatives and preserve polynomial structures, enhancing finite element analysis.

Contribution

It introduces a generalized bubble transform for differential forms that maintains key properties like commutation with exterior derivatives and polynomial preservation.

Findings

Decomposes differential forms into local bubbles on macroelements.

The transform commutes with the exterior derivative.

It is bounded in L^2 and preserves polynomial structures.

Abstract

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in a previous paper by the authors for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh of the domain, we build a map which decomposes piecewise smooth $k$-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of $k$-forms. Furthermore, the transform is bounded in $L^2$ and also on the appropriate subspace consisting of $k$-forms with exterior derivatives in $L^2$.