We survey recent contributions to finite element exterior calculus over manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations over manifolds. Our primary focus is on uniformly bounded commuting projections over 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 over 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 Cl\'ement or Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.

翻译：我们综述了在流形与曲面上开展有限元外微积分的最新贡献，并建立了一套统一的误差分析形式体系，用于处理流形上向量值偏微分方程。核心关注点是流形上一致有界的交换投影：此类投影将Sobolev de Rham复形映射到有限元de Rham复形，与微分算子可交换，且在Lebesgue范数下满足一致有界性。它们使得Hilbert复形的Galerkin理论能够适用于流形上的广泛内禀有限元方法。然而，这些内禀有限元方法通常不可计算，因此主要具有理论意义。由此引出我们的第二个要点：估计转向可计算近似问题时产生的几何变分偏差。最后，第三个要点阐述如何基于网格尺寸估计内禀有限元方法的逼近误差。若解不连续，则需借助修正的Clément或Scott-Zhang插值算子（这些算子支持分片的Bramble-Hilbert引理）实现此类估计。