An Approximate Bounded Cochain Projection

This paper presents a construction of a projector from an infinite-dimensional Hilbert complex of differential $k$-forms onto a finite-dimensional piecewise polynomial sub-complex. We demonstrate that, on contractable domains, the proposed projector attains the three properties of the Bounded Cochain Projector, namely that the projector is idempotent, uniformly bounded in the Sobolev norm, and it commutes with the exterior derivative. On non-contractible domains, the projector remains idempotent and uniformly bounded, while the commuting property is satisfied up to arbitrary accuracy.