Discrete Exterior Calculus

In this talk a view and vision will be given on the power of exterior calculus to model fundamental phenomena in physics. This will connect to an historical prospectives and to the fundamentals of why such techniques are far superior than what people still currently use. Specifically this will also address fluid structure interaction in a flying bird modeling and insights in gravity and quantum theory.

A novel formulation and numerical strategy for the simulation of geometrically nonlinear structures is presented. First, a non-canonical Hamiltonian (Poisson) formulation is introduced by including the dynamics of the stress tensor. This framework is developed for von-Kàrmàn nonlinearities in beams and plates, as well as geometrically nonlinear elasticity with Saint-Venant material behavior and applies to general geometric nonlinearities. In the case of plates, both negligible and non-negligible membrane inertia are considered. For the former case the two-dimensional elasticity complex is leveraged to express the dynamics in terms of the Airy stress function. The finite element discretization employs a mixed approach, combining a conforming approximation for displacement and velocity fields with a discontinuous stress tensor representation. A staggered, linear implicit time integration scheme is proposed, establishing connections with existing explicit-implicit energy-preserving methods. The stress degrees of freedom are statically condensed, reducing the computational complexity to solving a system with a positive definite matrix. The integration strategy preserves energy and angular momentum exactly. The methodology is validated through numerical experiments on the Duffing oscillator, a von-Kàrmàn beam, and a column undergoing finite deformations. Comparisons with fully implicit energy-preserving method and the leapfrog scheme demonstrate that the proposed approach achieves superior accuracy while maintaining energy stability. Additionally, it enables larger time steps compared to explicit schemes and exhibits computational efficiency comparable to the leapfrog method.

Discrete exterior calculus (DEC) is an important class of structure-preserving (mimetic, compatible) discretization method, with discrete analogues of key properties of the continuous exterior calculus such as the exact sequence property, integration by parts and the Hodge decomposition. This permits the design of numerical models with discrete analogues of fundamental physical properties such as conservation laws and involution constraints, along with strong guarantees on numerical stability. Additionally, unlike other SP discretizations such as finite element exterior calculus, it allows the representation of both straight and twisted differential forms. These are needed for certain physical theories, such as electrodynamics. However, conventional DEC is restricted to the case of scalar-valued differential forms (SVDFs). In this talk I will discuss an extension of DEC to the case of (vector) bundle-valued differential forms (BVDFs) in R^3, and demonstrate it's utility in designing a structure-preserving numerical model for compressible flow. If time permits, I will also discuss ideas towards extending DEC to BVDFs on arbitrary manifolds.

I will present our ongoing program at Princeton [1–7], which aims to develop a structure-preserving theoretical framework for modeling the dynamics of charge fields coupled to gauge fields. The focus is on capturing physical phenomena beyond the scope of perturbation theory in the radiation–matter coupling, and on enabling accurate derivation and numerical simulation of the resulting equations, both in the quantum regime and relevant semiclassical limits, using simplicial mesh-based numerical methods. Applications will be presented to some problems in scalar quantum electrodynamics, capturing the dynamics of low-lying collective excitations in superconducting quantum devices, and to the massive Schwinger Model, which describes the relativistic dynamics of charged leptons interacting with an electromagnetic gauge field in 1+1 dimensions. Within this framework, a DEC-inspired approach is used to derive an effective discretized field theory that describes the dynamics of spatially and/or temporally coarse-grained fields within a finite space-time volume, tailored to accurately capture the features resolvable by a physical measurement apparatus.[1] D. Pham et al., “Flux-Based Three-Dimensional Electrodynamic Modeling Approach to Superconducting Circuits and Materials.” Phys. Rev. A 107, 053704 (2023).[2] K. Sinha et al., “Radiative Properties of an Artificial Atom Coupled to a Josephson Junction Array.” Phys. Rev. A 106, 033714 (2022).[3] T. Maldonado et al., “Negative Electrohydrostatic Pressure Between Superconducting Bodies.” Physical Review B 110.1 (2024).[4] W. Fan et al., “Model Order Reduction for Open Quantum Systems Based on Measurement-Adapted Time-Coarse Graining.” arxiv:2410.23116[5] D. Pham et al., “Spectral Theory for Non-Linear Superconducting Microwave Systems: Extracting Relaxation Rates and Mode Hybridization.”[6] D. Pham et al., “Long-Time Soliton Dynamics via a Coarse-Grained Space-Time Method.", arxiv:2504.12286[7] T. Maldonado et al., “Mesoscopic Theory of the Josephson Junction.” Physical Review B 111, L140505 (2025).

This talk presents a versatile software framework for solving partial differential equations using the discrete exterior calculus (DEC) discretization method. Originating from our research group's development of a C++ library, the framework has been applied to a range of problems in acoustics, elastodynamics, electromagnetics, and quantum mechanics. Application areas include, for instance, wave propagation in photonic crystals and the evolution of Alice rings in spinor Bose-Einstein condensates. GPU acceleration has enabled efficient solutions of time-dependent Gross–Pitaevskii equations. Advanced features include space-time discretization in Minkowski space and higher-order Whitney form methods. More recently, development has focused on a general-purpose DEC solver implemented in Rust. Current work focuses on extending the library toward application areas such as plasma flows and nanoscale heat transport. The framework supports both time-dependent and time-harmonic problems, with the latter approached via the exact controllability method.

We provide a finite element discretization of $ell$-form-valued $k$-forms on triangulation in $mathbb{R}^{n}$ for general $k$, $ell$ and $n$ and any polynomial degree. The construction generalizes finite element Whitney forms for the de~Rham complex and their high-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension.

This talk discusses the development of a structure-preserving discretization of the exterior calculus of differential forms with values in a vector bundle over a combinatorial manifold equipped with a connection. Compared to their scalar-based counterparts which admit a well-established discretization via cochains, bundle-valued forms, (e.g., with values in the group of rotation matrices) present numerous difficulties when one tries to properly define a discrete counterpart to them and to the exterior covariant derivative operator acting on them. We show however that the use of specifically selected local frame fields allows for the construction of a discrete exterior covariant derivative of bundle-valued forms that not only satisfies the well-known Bianchi identities in this discrete realm, but also converges to its smooth equivalent under mesh refinement.

In this talk we use Whitney forms on the primal mesh and generalized Whitney forms developed by Christiansen on the dual mesh to write DEC only using forms.  This allows one to view DEC in a more FEEC-like setting. We then are able to use FEEC like analysis to give an error analysis for the Hodge Laplacian. It is important to note that for the zero Laplacian the analysis was already performed by Schulz and Tsogtgerel and we use some of their techniques to analyze all the Hodge Laplacians.  Our results only hold for uniformly well centered meshes which is quite limiting in practice. We are also able to prove superconvergence results when there is symmetry in the meshes. This is joint work with Pratyush Potu.

Differential complexes that arise as so-called BGG (Bernstein-Gelfand-Gelfand) sequences play an important role in several applications. The original constructions of such sequences rely on substantial background from algebra and/or differential geometry that is not easily accessible. In simplified constructions in the situation needed in applications therefore quite a bit of ad-hoc input is needed. Usually, this involves sums of de Rham complexes which are related by maps, which are "cleverly chosen" in such a way that the cohomology remains unchanged and at the same time they allow for passage to a subcomplex that computes the same cohomology. My talk is devoted to a simple instance of the BGG construction, in which the complex one considers is just the standard de Rham complex of a domain $U_R^3$ and in which the maps mentioned above arise in a more natural way. This is based on work of M. Rumin in the 1990's and is related to a branch of differential geometry known as contact geometry. The resulting complex has smaller spaces (pairs of functions instead of triples of functions in degree 1 and 2) and one second order operator in addition to two simple first order operators. In addition, it  is still invariant under an infinite dimensional group of diffeomorphisms. I'll outline the constructions in the case of a special contact form on $R^3$ that can be restricted to any domain $U$.

Discrete Exterior Calculus (DEC) has proven to be an effective modeling framework for computational physics on discrete spaces, particularly for complex multiphysics systems. This talk will present Decapodes.jl, a software package developed to facilitate the modeling and simulation of such systems using DEC. Decapodes provides a flexible framework for representing multiphysics systems on manifold-like discrete spaces in up to three dimensions, enabling researchers to encode, manipulate, and analyze complex models algorithmically. We will discuss how the DEC plays a role in the automatic assembly of multiphysics implementations from high-level physics specifications. Recent developments support advanced fluid mechanical models, including pressure projection techniques for Euler’s equations, and vorticity formulations of the incompressible Navier-Stokes equations. Capabilities of the software package are demonstrated through illustrative examples and highlight its applications in mathematical physics, systems theory, and scientific computing. The talk will highlight open opportunities to bring the DEC to more practitioners through software development.

When second-order boundary value problems are solved in finite-dimensional spaces, it is inevitable that some of the properties inherent to the corresponding continuous problem must be compromised. Although the literature on numerical techniques is extensive, relatively little attention has been paid to how the essential structural properties are preserved in finite-dimensional models. A reason for this is historical; The development of numerical techniques has been driven by the needs of various fields of physics, and hence, there has been less emphasis on the structures boundary value problems share and on their preservation on finite dimensional settings. In this work, we adopt a complementary perspective by first asking: what do second-order boundary value problems have in common? To address this, we begin by defining a general class of such problems and subsequently demonstrate how well-known instances can be derived from this framework. We then transition to the finite-dimensional setting, focusing on the approximations involved. As a result, we observe that widely used numerical techniques—such as the finite element and finite difference methods—are, in essence, variations of a common underlying approach rather than fundamentally distinct methods