Smoothed projections over manifolds in finite element exterior calculus

We develop commuting finite element projections on compact Riemannian smooth manifolds. The commuting projections are constructed using localized smoothing operators, building on a classical construction by de~Rham. The projections map onto finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. They are uniformly bounded on Lebesgue spaces of differential forms over shape-regular families of triangulations. These bounded cochain projections are a key component in the stability and convergence analysis of finite element methods for the Hodge--Laplace equation on manifolds. We derive Galerkin error estimates for the intrinsic finite element method and prove a piecewise Bramble--Hilbert lemma. Since many practical computations use extrinsic finite element methods on approximate computational manifolds, we also analyze the resulting geometric error. We bound the geometric variational crime in terms of the discrepancy between the exact Riemannian metric and its computational approximation.