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 Cl\'ement or Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.

翻译：我们综述了流形与曲面上有限元外微积分的最新贡献，构建了一个用于流形上向量值偏微分方程误差分析的综合形式体系。主要关注流形上一致有界的交换投影：这些投影将Sobolev de Rham复形映射至有限元de Rham复形，与微分算子交换，且满足Lebesgue范数下的一致有界性。它们使得希尔伯特复形的伽辽金理论能够适用于流形上大范围的内蕴有限元方法。然而，这些内蕴有限元方法通常不可计算，因而主要具有理论意义。这引出了我们的第二个要点：评估转向可计算近似问题时产生的几何变分误差。最后，第三个要点阐述如何通过网格尺寸估计内蕴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（通过断裂Bramble-Hilbert引理）实现此类估计。