We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula. This theorem is the main mathematical result of the paper.

我们从电气工程的角度来看待计算物理，并提出一些在计算物理文献中尚未确立的数学概念，它们本身就是该领域的机遇。我们讨论椭圆复形并强调范畴理论背景及其作为代数拓扑、微分几何和建模软件设计之间的统一语言的作用。特别是，无处不在的自然概念是核心。自然微分算子在三角流形的上链上具有函数类似物。为了建立这种对应关系，我们推导了涉及单纯形和重心坐标的公式，定义了离散向量场和离散李导数，作为嘉当魔术公式的离散模拟的结果。该定理是本文的主要数学结果。