A Discrete Exterior Calculus of Bundle-valued Forms

TL;DR

This work develops a convergent, structure-preserving discrete exterior calculus for bundle-valued forms by introducing parallel-propagated frame fields (PPFs) and a de Rham/Whitney-like sampling for bundle-valued data. The authors define discrete connections, curvature, and a discrete exterior covariant derivative d^∇, incorporating an alternation (averaging) step to achieve higher-order accuracy. They prove that the discrete operators satisfy naturality, antisymmetry, and both differential and algebraic Bianchi identities, and provide extensive numerical tests confirming convergence under mesh refinement for vector- and endomorphism-valued forms, including 1- and 2-forms. The framework enables faithful discretization of geometric structures central to physics (e.g., Yang–Mills, general relativity) and paves the way for topology-preserving simulations in higher dimensions.

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.

Figures (21)

• Figure 1: Discrete Connections. A discrete vector bundle $\boldsymbol{E}$ assigns a vector space to each vertex on a mesh. To compare quantities in different fibers, we need to encode how the individual fibers are connected. A connection consists of a collection of linear maps between neighboring fibers. In the figure, each box represents a frame for each fiber space over a vertex. The right part of the figure visualizes the local action of two different discrete connections. Starting with an initial vector $v \in \boldsymbol{E}_{p_0}$ (gray), we illustrate the different parallel transports induced by two different connections, $\nabla^1$ and $\nabla^2$. Different connections on the same vector bundle result in distinct notions of parallel transport and curvature.
• Figure 2: Curvatures. Illustration of the discrete curvatures for different cut vertices.
• Figure 3: We investigate two distinct frame fields on the same vector bundle with an identical connection $(E, \nabla)$. In this context, the boxes represent the individual fibers of the bundle. Both bundles exhibit a similar "twisted" structure, whereby the fibers are equally interconnected, implying that the connection (or parallel transport) of the bundle remains consistent. With a local frame construction facilitated by a rotation field $R^{\nabla}$, we can alter both the differential and the connection $1-$form. By employing a parallel-propagated frame field (left), we achieve ${\omega}^{\nabla}(c) = 0$ at a specific point $c$. In contrast, when using a non-parallel-propagated frame (right), the connection $1-$form does not vanish at $c$. Intuitively, the parallel-propagated frame field on the left "follows" the bundle along radial lines emanating from $c$. Consequently, $\Tilde{\omega}$ vanishes along these radial lines. This behavior highlights the distinctive properties of the two frame fields in terms of their interaction with the bundle's connection.
• Figure 4: Retraction. In this illustration, we consider a simplicial cell $\sigma$ and demonstrate the retraction of $\sigma$ onto one of its vertices $v \!\in\! \sigma$. To aid visualization, an offset from the initial triangle is used to depict the shrinkage induced by the retraction function $\varphi$. For any point $p \!\in\! \sigma$, the aforementioned retraction induces radial joining paths from $p$ to $v$, which can serve as paths for parallel transport from source point $p$ to $v$. This parallel transport, defined along these paths, can be used subsequently to define the parallel propagated frame based in $v$.
• Figure 5: Retraction-based parameterization. (Left): a retraction of the disk defines a $(\rho,\theta)$ (polar) parameterization of the (punctured) disk, where the $\rho$ direction is along the retraction direction, while the $\theta$ direction corresponds to a transversal direction. (Right): Starting from a parallel-transported frame field $F_\text{eval}$ from $v_\text{eval}$ along the boundary, we can parallel-propagate this boundary frame field towards the interior of the punctured disk along the curves corresponding to the $y$ direction of this retraction-based parameterization. The curvature integral simplifies to the discrepancy between frames at $v_{\text{cut}}$, transported from $v_{\text{eval}}$ via the retraction, and evaluated in $\operatorname{Hom} (\mathbf{E}_{v_\text{cut}},\mathbf{E}_{v_\text{eval}})$.

Theorems & Definitions (53)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4: Bianchi Identities in (Pseudo) Riemannian Geometry
• Definition 3.1: Discrete Vector Bundle
• Definition 3.2: Section of Discrete Frame Bundle
• Definition 3.3: Discrete Connection
• Definition 3.4: Pullback Connection Hirani_Bianchi
• Definition 3.5: Discrete (1,0)-tensor-valued $\ell-$form
• Definition 3.6: Discrete $(1,1)-$tensor-valued $\ell-$form