Local Finite Element Approximation of Sobolev Differential Forms

TL;DR

This paper develops new finite element interpolation techniques for Sobolev differential forms, providing local error estimates and minimal smoothness requirements within the finite element exterior calculus framework.

Contribution

It generalizes classical interpolants to differential forms and derives a broken Bramble-Hilbert Lemma, enhancing approximation theory for vector-valued finite element methods.

Findings

Generalized Clément and Scott-Zhang interpolants for differential forms

Derived local error estimates based on mesh size

Applicable to curl- and divergence-conforming finite element methods

Abstract

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Cl\'ement interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.