We present some aspects of the theory of finite element exterior calculus as applied to partial differential equations on manifolds, especially manifolds endowed with an approximate metric called a Regge metric. Our treatment is intrinsic, avoiding wherever possible the use of preferred coordinates or a preferred embedding into an ambient space, which presents some challenges but also conceptual and possibly computational advantages. As an application, we analyze and implement a method for computing an approximate Levi-Civita connection form for a disc whose metric is itself approximate.

翻译：本文阐述了有限元外微积分理论在流形上偏微分方程中的应用，特别关注于配备近似度量（称为Regge度量）的流形。我们的处理方式是内蕴的，尽可能避免使用特定坐标或预设的嵌入环境空间，这带来了一些挑战，但也具有概念上及潜在计算上的优势。作为应用，我们分析并实现了一种计算方法，用于计算具有近似度量的圆盘的近似Levi-Civita联络形式。