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 Clément 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.
Highlights
With this article we contribute to an aspect of vector-valued finite element methods which has seen increasing interest throughout recent years, namely the detailed study of quantitative approximation estimates
We present our results in the framework of finite element exterior calculus (FEEC)
Using Veeser’s exposition [40] as a primary source, we prove a broken Bramble-Hilbert lemma for finite element differential forms
Summary
With this article we contribute to an aspect of vector-valued finite element methods which has seen increasing interest throughout recent years, namely the detailed study of quantitative approximation estimates. Our Scott-Zhang-type interpolant uses only momenta over full-dimensional cells and facets Prospective applications of this broken Bramble-Hilbert lemma include the convergence theory of finite element exterior calculus over surfaces and manifolds. We mention the quasi-optimal interpolant of Ern and Guermond [18] as the apparently first such construction in the literature Their projection operator, which generalizes ideas of Oswald [35] to curl- and divergence-conforming finite element spaces, satisfies similar local error estimates as the Clement interpolant and can be modified to satisfy homogeneous boundary conditions. Apart from quasi-interpolation error estimates for vector-valued finite element methods, for which we study the Clement interpolant and the Scott-Zhang interpolant in finite element exterior calculus, we are interested in what has been in circulation as broken Bramble-Hilbert lemma in recent years.
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