We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.

我们开发了一种有限元系统理论，目的是对向量丛的部分进行离散化，特别是在弹性理论中出现的部分。在存在曲率的情况下，我们证明了离散的 Bianchi 恒等式。在平面情况下，我们证明了一个关于上同调群的 de Rham 定理。我们检查了一些已知的弹性应力-位移公式的混合有限元是否适合我们的框架。我们还在二维中定义了具有良好线性曲率的度量的第一个符合有限元空间，对应于具有圣维南相容条件的应变张量。具有刚性运动系数的 Cochains 在关联连续和离散弹性复合体中起关键作用。