De Rham Complex Research Articles

Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex

Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

$q$–de Rham complexes of higher level

In this article, we construct two kinds of de Rham–like complexes which compute the cohomology of complete crystals on the higher-level q -crystalline site, which was introduced in a previous article by the author. One complex is the q -analog of the higher de Rham complex constructed by Miyatani, and another complex is the q -analog of the jet complex constructed by Le Stum–Quirós. The complexes we construct can also be regarded as the higher-level analogs of the q –de Rham complex.

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra K is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of K⊗K¯ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in C3 this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of C3. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

Vanishing of local cohomology with applications to Hodge theory

Let H=((H,F•),L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M∈HM(U). Let X be a compactification of U, and j:U↪X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M∈DbMHM(X). In particular, we study the graded pieces of the de Rham complex GrpFDR(j+M)∈Dcohb(X), and the Hodge structure of Hi(U,L) for i in sufficiently low degrees.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Yang–Mills models on Noncommutative background

In our previous publications we have developed some elements of Noncommutative calculus on the enveloping algebras of Am type, in particular, analogs of the partial derivatives and de Rham complex were defined. Also, we introduced the notion of quantization with Noncommutative configuration space and quantized a few dynamical models in this sense. In the current paper we propose a method of quantizing the Yang–Mills models in the same sense.

Cartan calculi on the free loop spaces

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André–Quillen cohomology of a commutative differential graded algebra A on the Hochschild homology of A in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space M to the derivation ring on the loop cohomology of M. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the Γ1 map due to Félix and Thomas.

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations

The Hilbert spaces H(curl) and H(div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin’s criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H1-conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H1-conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model.

Correction to: Hypercohomologies of truncated twisted holomorphic de Rham complexes

Correction to: Hypercohomologies of truncated twisted holomorphic de Rham complexes

Hyperplane Arrangements Satisfy (Un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr {D}_{X}$ -modules

Abstract For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$ -module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem

Abstract In this work, following the discrete de Rham approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete set of Poincaré-type inequalities. The discrete complex is then used to derive a novel discretization method for a quad-rot problem, which, unlike other schemes in the literature, does not require the forcing term to be prepared. We carry out complete stability and convergence analyses for the proposed scheme and provide numerical validation of the results.

Geometric Hodge filtered complex cobordism

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration.

BGG Sequences with Weak Regularity and Applications

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document}. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation

In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

De Rham compatible Deep Neural Network FEM

On general regular simplicial partitions T of bounded polytopal domains Ω⊂Rd, d∈{2,3}, we construct exact neural network (NN) emulations of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical “Raviart–Thomas element”, and the “Nédélec edge element”. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions T of Ω are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension d≥2. Our “FE-Nets” are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra Ω⊂R3. They are thus an essential ingredient in the application of e.g., the methodology of “physics-informed NNs” or “deep Ritz methods” to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the “Crouzeix–Raviart” elements and Hybridized, Higher Order (HHO) methods.

Finite element de Rham and Stokes complexes in three dimensions

Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H ( div ) H(\operatorname {div}) -conforming finite elements and H ( curl ) H(\operatorname {curl}) -conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H ( div ) H(\operatorname {div}) -conforming finite elements is established, and the exactness of these finite element complexes is proven.