  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.
