Averaging-based local projections in finite element exterior calculus

Abstract

We construct local projection operators onto spaces of finite element differential forms on simplicial meshes. The projections are locally stable in Lebesgue and Sobolev--Slobodeckij norms, uniformly with respect to the mesh size, enforce homogeneous partial boundary conditions, and satisfy piecewise Bramble--Hilbert estimates localized around element stars. Special cases include the Ern--Guermond projection and a modified Clément-type interpolant that is idempotent. These results lead to an equivalence between global and local best-approximation errors for curl- or divergence-conforming finite element spaces. Our construction combines techniques for the Ern--Guermond and the Scott--Zhang projections within the framework of finite element exterior calculus, and addresses three different regularity regimes. We instantiate the abstract projection for the divergence-conforming Brezzi--Douglas--Marini and Raviart--Thomas elements and for the curl-conforming Nédélec elements. As an auxiliary result of independent merit, we contribute an extension operator for differential forms that preserves higher Sobolev--Slobodeckij regularity.

arXiv comment36 pages. Revised. Submitted