The discretization of Cartan's exterior calculus of differential forms has been fruitful in a variety of theoretical and practical endeavors: from computational electromagnetics to the development of Finite-Element Exterior Calculus, the development of structure-preserving numerical tools satisfying exact discrete equivalents to Stokes' theorem or the de Rham complex for the exterior derivative have found numerous applications in computational physics. However, there has been a dearth of effort in establishing a more general discrete calculus, this time for differential forms with values in vector bundles over a combinatorial manifold equipped with a connection. In this work, we propose a discretization of the exterior covariant derivative of bundle-valued differential forms. We demonstrate that our discrete operator mimics its continuous counterpart, satisfies the Bianchi identities on simplicial cells, and contrary to previous attempts at its discretization, ensures numerical convergence to its exact evaluation with mesh refinement under mild assumptions.

翻译：卡坦微分形式外微分的离散化已在众多理论与应用领域取得丰硕成果：从计算电磁学到有限元外微积分的发展，满足斯托克斯定理精确离散等价形式或外微分算子的德拉姆复形结构保持数值工具的构建，已在计算物理学中获得广泛应用。然而，针对配备联络的组合流形上向量丛取值的微分形式，建立更广义离散微积分的研究尚显不足。本工作提出向量丛取值微分形式外协变导数的离散化方案。我们证明该离散算子能模拟其连续对应物，在单纯形胞腔上满足比安基恒等式，且与既往离散化尝试不同，在温和假设条件下能通过网格细化确保数值解向精确值的收敛。