Convergence of finite element exterior calculus for the Hodge-Dirac and Hodge-Laplace problems on closed Riemannian manifolds

We prove that the finite element exterior calculus (FEEC) approximation to the Hodge-Dirac and Hodge-Laplace problems on any closed oriented Riemannian manifold converges. Its approximate solutions, together with their exterior derivatives, converge to the continuum solutions for every source, and at the rate of the mesh size when the source and its codifferential are square integrable. We use Licht's estimate for the FEEC approximation of forms with one square-integrable derivative by finite element forms. We combine this with a new abstract error bound for approximations of Hilbert complexes. This bound uses a version of Di Pietro and Droniou's third Strang lemma. To apply these estimates, we show that every closed manifold admits a family of arbitrarily fine triangulations whose shapes stay uniformly under control.