Calculus Of Differential Forms Research Articles

Trefftz co-chain calculus

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on {mathbb {R}}^n. In the spirit of domain decomposition, we partition {mathbb {R}}^n=Omega cup Gamma cup Omega _{+}, Omega a bounded Lipschitz polyhedron, Gamma :=partial Omega , and Omega _{+} unbounded. In Omega , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Omega _{+}, we rely on a meshless Trefftz–Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Gamma . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.

Polynomial Finite-Size Shape Functions for Electromagnetic Particle-in-Cell Algorithms Based on Unstructured Meshes

In this article, we describe a novel implementation of finite-size (super) particles described by polynomial-based spatial shape factors in electromagnetic particle-in-cell (EM-PIC) algorithms for kinetic plasma simulations based on unstructured meshes. The proposed implementation is aimed at mitigating (spurious) numerical Cherenkov radiation effects while preserving exact charge conservation. This is achieved by employing a representation of Maxwell's equations based on the exterior calculus of differential forms and a discretization of twisted differential forms associated with the primal mesh by Whitney forms, while ordinary differential forms are associated with the dual mesh. The set of full-discrete Maxwell's equations is then obtained by using a finite-element Galerkin method and a leap-frog time integration, leading to a mixed D-H finite-element time-domain Maxwell field solver. In the proposed EM-PIC implementation, the gather and scatter steps of the algorithm yield integrals of polynomials over polyhedral domains in 3-D space (or polygonal domains in 2-D space) that can be evaluated analytically. We provide numerical examples confirming charge conservation down to machine precision on an unstructured mesh.

Differential forms on locally convex spaces and the stokes formula

In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS). The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first author. Moreover, on some subspace of differential forms of the Sobolev type with respect to a differentiable measure we establish a formula expressing the operator adjoint to the exterior differential via standard operations of the calculus of differential forms and the logarithmic derivative. This connection was established earlier under stronger constraints imposed either on the LCS or on the measure, or on differential forms (the smoothness condition).

Comparison of discrete exterior calculus and discrete-dipole approximation for electromagnetic scattering

In this study, we perform numerical computations of electromagnetic fields including applications such as light scattering. We present two methods for discretizing the computational domain. One is the discrete-dipole approximation (DDA), which is a well-known technique in the context of light scattering. The other approach is the discrete exterior calculus (DEC) providing the properties and the calculus of differential forms in a natural way at the discretization stage. Numerical experiments show that the DEC provides a promising alternative for solving the general Maxwell equations.

Evolution of helicity in fluid flows

An invariant helicity integral and a differential helicity evolution equation are found for viscous fluid flows. A geometrodynamical approach is used, which includes a vortex field. The vortex field is derivable from a vector potential A. The vector potential is then used to characterize the evolution of flow topology. The source of the helicity is found to be the topological parity k=2λω⋅ζ and the moving boundary surfaces of the fluid. Here, ω and ζ are the vorticity and swirl components of the vortex field {ω,ζ} and λ is a constitutive or material parameter of the fluid. Our first result using the vector calculus identifies the scalar helicity as ht=Aω. This result is then generalized using the calculus of differential forms, yielding other results including the existence of a helicity current vector proportional to (ϕω−λA×ζ).

New Perspectives on the Kähler Calculus and Wave Functions

In 1960-1962, Kahler produced a generalization of Cartan’s calculus of differential forms which he claimed to be a common language for relativity and quantum mechanics. It revolves around the “Kahler-Dirac (KD) equation”, which plays a role in this calculus comparable to that of the equations of structure in geometry. Its reach is such that the input differential form needed to solve the relativistic hydrogen atom is just a very simple one among the possible inputs.

Finite elements in computational electromagnetism

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

Discrete Hodge operators

Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.

Triangle lattice Green functions for vector fields

All the identities and integral theorems of vector calculus are contained in the calculus of differential forms. The analogy between the exterior calculus of forms and the homology theory of a cell complex yields discrete lattice models for an array of interesting physical phenomena. These models, based on arbitrary combinations of coupled scalar and polar or axial vector field quantities, can be manipulated as conveniently as the standard scalar tight-binding models. We develop Green functions (GFs) for an infinite hierarchy of such models expressed using the four fundamental operators of the triangle lattice. The triangle lattice is distinguished among two-dimensional grid types for having the highest possible isotropy. Closed formulae are derived for GFs of the scalar and vector models that belong to a hierarchy generated by the four fundamental operators of the triangle family of lattices. The particular example of lattice electromagnetism coupled to an elastic distortion field is treated in detail. Topological properties not dependent upon symmetry split the response functions into plasmon- and polariton-like parts. Since the fundamental vector operators of the triangle lattice are related simply to adjacency matrices of the Kagome lattice, scalar GFs for this lattice are found also as a byproduct.

Discrete Hodge-Operators: An Algebraic Perspective

Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metric- dependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms.

Discrete Hodge-Operators: an Algebraic Perspective - Abstract

Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metric-dependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms.

HIGHER ORDER WHITNEY FORMS - Abstract

The calculus of differential forms can be used to devise a unified description of discrete differential forms of any order and polynomial degree on simplicial meshes in any spatial dimension. A general formula for suitable degrees of freedom is also available. Fundamental properties of nodal interpolation can be established easily. It turns out that higher order spaces, including variants with locally varying polynomial order, emerge from the usual Whitney-forms by local augmentation. This paves the way for an adaptive p-version approach to discrete differential forms.

Teaching electromagnetic field theory using differential forms

The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory. First, films clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Ampere's and Faraday's laws obtain graphical representations that are as intuitive as the representation of Gauss's law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations. We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.

Noncommutative Geometry and Integrable Models

A construction of conservation laws for σ-models in two dimensions is generalized in the framework of noncommutative geometry of commutative algebras. This is done by replacing theordinary calculus of differential forms with other differentialcalculi and introducing an analogue of the Hodge operator on thelatter. The general method is illustrated with several examples.

The Dirac-Kähler equation and fermions on the lattice

The geometrical description of spinor fields by E. Kahler is used to formulate a consistent lattice approximation of fermions. The relation to free simple Dirac fields as well as to Susskind's description of lattice fermions is clarified. The first steps towards a quantized interacting theory are given. The correspondence between the calculus of differential forms and concepts of algebraic topology is shown to be a useful method for a completely analogous treatment of the problems in the continuum and on the lattice.

The differential geometric structure of general mechanical systems from the Lagrangian point of view

Several differential-geometric points of view on analytical mechanics of systems with a finite number of degrees of freedom are developed in generality, emphasizing Cartan’s calculus of differential forms and Ehresmann’s theory of jet spaces. The classical theory of Lagrange’s equations with external forces and constraints (‘‘holonomic’’ or ‘‘nonholonomic’’) is put into an invariant and coordinate-free form. The relation between this ‘‘Lagrangian’’ and the ‘‘Hamiltonian-symplectic’’ approach, which is that most extensively used in the contemporary mathematical physics literature, is also developed.

On the concept of divergence

This paper presents a motivated introduction to the concept of divergence of a vector field in Euclidean space at an advanced level. Use is made of some basic concepts from the exterior algebra of k‐forms and some from the calculus of differential forms.

The Geometric Approach to Sets of Ordinary Differential Equations and Hamiltonian Dynamics

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.