Towards Finite Element Exterior Calculus on Manifolds: Commuting Projections, Geometric Variational Crimes, and Approximation Errors

We survey recent contributions to finite element exterior calculus on manifolds and surfaces, giving a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. We first discuss uniformly bounded commuting projections that map Sobolev de Rham complexes on manifolds onto finite element de Rham complexes, commute with the differential operators, and satisfy uniform bounds in Lebesgue norms. They enable the Galerkin theory of Hilbert complexes for numerous intrinsic finite element methods on manifolds. However, these first intrinsic finite element methods are generally not computable and thus of theoretical interest, which leads to our second point: estimating the geometric variational crime incurred by transitioning to computable approximate problems. Thirdly, we estimate the approximation error of the intrinsic finite element method in terms of the mesh size. If the solution is not continuous, then the error estimate uses modified Clément or Scott–Zhang interpolants that facilitate a broken Bramble–Hilbert lemma.