Towards finite element exterior calculus on manifolds: commuting projections, geometric variational crimes, and approximation errors

TL;DR

This work develops the finite element exterior calculus framework for PDEs on manifolds by constructing uniformly bounded commuting projections from the $L^2$ de Rham complex to finite element de Rham complexes, enabling stable, quasi-optimal Galerkin discretizations on manifolds. It systematically analyzes the gap between ideal intrinsic finite element methods (which are often noncomputable) and computable approximations through a geometric variational crime framework, quantifying the impact of geometry and metric approximations on solution accuracy. The paper also establishes how to bound the intrinsic approximation error in terms of mesh size using broken Bramble–Hilbert techniques and generalized interpolation operators suitable for low-regularity solutions. Together, these results provide a rigorous pathway for error control in vector-valued PDEs on surfaces and manifolds, with practical implications for computations in physics and engineering on curved geometries.

Abstract

We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on uniformly bounded commuting projections on manifolds: these projections map from Sobolev de Rham complexes 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 a large range of intrinsic finite element methods on manifolds. However, these intrinsic finite element methods are generally not computable and thus primarily of theoretical interest. This leads to our second point: estimating the geometric variational crime incurred by transitioning to computable approximate problems. Lastly, our third point addresses how to estimate the approximation error of the intrinsic finite element method in terms of the mesh size. If the solution is not continuous, then such an estimate is achieved via modified Clément or Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.