Local $L^2$-bounded commuting projections in FEEC

TL;DR

This paper introduces local $L^2$-bounded projections in finite element exterior calculus that commute with the exterior derivative, enhancing the mathematical tools available for finite element methods.

Contribution

It presents the first construction of local projections into FEEC spaces that are both $L^2$-bounded and commute with the exterior derivative.

Findings

Projections are bounded in $L^2$ norm.

Projections commute with the exterior derivative.

Applicable to canonical FEEC spaces.

Abstract

We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in $L^2$ and commute with the exterior derivative.