We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on manifolds. The commuting projections use localized mollification operators, building upon a classical construction by de Rham. These projections are uniformly bounded on Lebesgue spaces of differential forms and map onto intrinsic finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. We analyze the Galerkin approximation error. Since practical computations use extrinsic finite element methods over approximate computational manifolds, we also analyze the geometric error incurred.

翻译：我们发展了在光滑黎曼流形上具有交换性的有限元投影。这一对有限元外微积分的拓展，建立了流形上Hodge-Laplace方程有限元方法的稳定性和收敛性。该交换性投影采用基于de Rham经典构造的局部磨光算子，在微分形式的勒贝格空间上一致有界，并映射到依据流形内在光滑三角剖分定义的固有有限元空间。我们分析了伽辽金逼近误差。由于实际计算中常采用近似计算流形上的外在有限元方法，本文还分析了由此产生的几何误差。