A Discrete Exterior Calculus of Bundle-valued Forms

TL;DR

This paper introduces a novel discrete exterior calculus for bundle-valued differential forms, ensuring convergence and preserving geometric identities, thereby extending computational tools in physics and geometry.

Contribution

It develops a discretization of the exterior covariant derivative for bundle-valued forms that mimics continuous properties and guarantees convergence, filling a gap in discrete differential geometry.

Findings

Discrete operator satisfies Bianchi identities on simplicial cells

Ensures numerical convergence with mesh refinement

Extends exterior calculus to bundle-valued forms

Abstract

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.