Averaging-based local projections in finite element exterior calculus

TL;DR

This paper introduces a new projection operator for finite element differential forms that is locally bounded, incorporates boundary conditions, and improves approximation estimates within the finite element exterior calculus framework.

Contribution

It develops a novel projection operator with enhanced properties, including a new modified Clément-type interpolant, applicable to various finite element differential forms.

Findings

Projection is locally bounded in Lebesgue and Sobolev norms.

Enables equivalence between local and global best approximations.

Applicable to BDM, Nédélec, and Raviart-Thomas elements.

Abstract

We develop projection operators onto finite element differential forms over simplicial meshes. Our projection is locally bounded in Lebesgue and Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover, it incorporates homogeneous boundary conditions and satisfies a local broken Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond projection and a modified Cl\'ement-type interpolant with the projection property. The latter seems to be a new result even for Lagrange elements. This projection operator immediately enables an equivalence result on local- and global-best approximations. We combine techniques for the Scott-Zhang and Ern-Guermond projections and adopt the framework of finite element exterior calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini, N\'ed\'elec, and Raviart-Thomas elements.