arXiv subjects

Yakov Berchenko-Kogan

Publications and source records attributed to Yakov Berchenko-Kogan.

15 recordsLinked to original sources

arXiv2025

Finite Element Spaces of Double Two-Forms With Polynomial Coefficients

We develop finite element spaces of symmetric tensor products of two-forms with polynomial coefficients. In three dimensions, these give higher order finite element spaces of matrix fields with normal-normal continuity, which have applications to the TDNNS method for elasticity, for example. In general dimension, these spaces can be used to represent the Riemann curvature tensor in numerical relativity. In many ways, our methods parallel Li's work generalizing Regge calculus to higher order, as Regge elements can be thought of as symmetric tensor products of one-forms. However, whereas the constant coefficient Regge space has one shape function per edge, the constant coefficient space of double-forms in our paper has one shape function per triangle and two shape functions per tetrahedron, so we must address the fact that there are shape functions of two different types. Like Li, we obtain an explicit geometrically decomposed basis of shape functions.

math.NA↗

arXiv2025

Finite element spaces of double forms

The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or $(p,q)$-forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of $(p,q)$-forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor.

math.NA↗

arXiv2025

Extension operators and geometric decompositions

Geometric decomposition is a widely used tool for constructing local bases for finite element spaces. For finite element spaces of differential forms on simplicial meshes, Arnold, Falk, and Winther showed that geometric decompositions can be constructed from extension operators satisfying certain properties. In this paper, we generalize their results to function spaces and meshes satisfying very minimal hypotheses, while at the same time reducing the conditions that must hold for the extension operators. In particular, the geometry of the mesh and the mesh elements can be completely arbitrary, and the function spaces need only have well-defined restrictions to subelements. In this general context, we show that extension operators yield geometric decompositions for both the primal and dual function spaces. Later, we specialize to simplicial meshes, and we show that, to obtain geometric decompositions, one needs only to construct extension operators on the reference simplex in each dimension. In particular, for simplicial meshes, the existence of geometric decompositions depends only on the dimension of the mesh.

math.NA↗

arXiv2024

Blow-up Whitney forms, shadow forms, and Poisson processes

The Whitney forms on a simplex $T$ admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of $T$. Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow $k$-forms that are well-suited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the $k$-dimensional faces of the blow-up $\tilde T$ of the simplex $T$. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of $\tilde T$, which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation.

math.NA↗

arXiv2022

Finite element approximation of the Levi-Civita connection and its curvature in two dimensions

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes.

math.NA↗

arXiv2021

Symmetric bases for finite element exterior calculus spaces

In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart--Thomas, Brezzi--Douglas--Marini, and N\'ed\'elec finite element spaces for simplicial triangulations. In a recent paper, Licht asks whether, on a single simplex, one can construct bases for these spaces that are invariant with respect to permuting the vertices of the simplex. For scalar fields, standard bases all have this symmetry property, but for vector fields, this question is more complicated: such invariant bases may or may not exist, depending on the polynomial degree of the element. In dimensions two and three, Licht constructs such invariant bases for certain values of the polynomial degree $r$, and he conjectures that his list is complete, that is, that no such basis exists for other values of $r$. In this paper, we show that Licht's conjecture is true in dimension two. However, in dimension three, we show that Licht's ideas can be extended to give invariant bases for many more values of $r$; we then show that this new larger list is complete. Along the way, we develop a more general framework for the geometric decomposition ideas of Arnold, Falk, and Winther.

math.NA↗

arXiv2021

Charge-conserving hybrid methods for the Yang-Mills equations

The Yang-Mills equations generalize Maxwell's equations to nonabelian gauge groups, and a quantity analogous to charge is locally conserved by the nonlinear time evolution. Christiansen and Winther observed that, in the nonabelian case, the Galerkin method with Lie algebra-valued finite element differential forms appears to conserve charge globally but not locally, not even in a weak sense. We introduce a new hybridization of this method, give an alternative expression for the numerical charge in terms of the hybrid variables, and show that a local, per-element charge conservation law automatically holds.

math.NA↗

arXiv2021

Duality in finite element exterior calculus and Hodge duality on the sphere

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite element exterior calculus began with the $\mathcal P_rΛ^k$ and $\mathcal P_r^-Λ^k$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal P_rΛ^k$ and $\mathring{\mathcal P}_{r+k+1}^-Λ^{n-k}$ and between $\mathcal P_r^-Λ^k$ and $\mathring{\mathcal P}_{r+k}Λ^{n-k}$. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard $n$-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $\mathcal P_rΛ^k$, $\mathcal P_r^-Λ^k$, $\mathring{\mathcal P}_rΛ^k$ and $\mathring{\mathcal P}_r^-Λ^k$ spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $\mathcal P_rΛ^k$ and $\mathring{\mathcal P}_{r+k+1}^-Λ^{n-k}$ and between $\mathcal P_r^-Λ^k$ and $\mathring{\mathcal P}_{r+k}Λ^{n-k}$, which we illustrate with examples.

math.NA↗

arXiv2020

Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly $-1$. Finally, we present some preliminary results in higher dimensions and six potential directions for future work.

math.DG↗

arXiv2020

Numerically computing the index of mean curvature flow self-shrinkers

Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropriate functional. In this paper, we present a numerical method for computing the index of rotationally symmetric self-shrinkers. We apply this method to the Angenent torus, the first known nontrivial example of a self-shrinker. We find that, excluding dilations and translations, the index of the Angenent torus is $5$, which is consistent with the lower bound of $3$ from the work of Liu and the upper bound of $29$ from our earlier work. Also, we unexpectedly discover two additional variations of the Angenent torus with eigenvalue $-1$.

math.DG↗

arXiv2019

The entropy of the Angenent torus is approximately 1.85122

To study the singularities that appear in mean curvature flow, one must understand self-shrinkers, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In 1989, Angenent constructed the first nontrivial example of a self-shrinker, a torus. A key quantity in the study of the formation of singularities is the entropy, defined by Colding and Minicozzi based on work of Huisken. The values of the entropy of spheres and cylinders have explicit formulas, but there is no known formula for the entropy of the Angenent torus. In this work, we numerically estimate the entropy of the Angenent torus using the discrete Euler-Lagrange equations.

math.DG↗

arXiv2019

Constraint-preserving hybrid finite element methods for Maxwell's equations

Maxwell's equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively. However, one or both of these constraints may be violated when one applies a finite element method to discretize in space. This is a well-known and longstanding problem in computational electromagnetics. We use domain decomposition to construct a family of primal hybrid finite element methods for Maxwell's equations, where the Lagrange multipliers are shown to correspond to a numerical trace of the magnetic field and a numerical flux of the electric flux density. Expressing the charge-conservation constraint in terms of this numerical flux, we show that both constraints are strongly preserved. As a special case, these methods include a hybridized version of N\'ed\'elec's method, implying that it preserves the constraints more strongly than previously recognized. These constraint-preserving properties are illustrated using numerical experiments in both the time domain and frequency domain. Additionally, we observe a superconvergence phenomenon, where hybrid post-processing yields an improved estimate of the magnetic field.

math.NA↗

arXiv2018

Duality in finite element exterior calculus

In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex $T$, the $\mathcal P_r\Lambda^k(T)$ spaces and the $\mathcal P_r^-\Lambda^k(T)$ spaces, where $k$ is the degree of the form and $r$ is the degree of its coefficients. The geometric decomposition for these finite element spaces hinges on a duality relationship between the $\mathcal P$ and $\mathcal P^-$ spaces proved by Arnold, Falk, and Winther. In this article, we give a natural alternate construction of the $\mathcal P_r\Lambda^k(T)$ and $\mathcal P_r^-\Lambda^k(T)$ spaces, leading to a new basis-free proof of this duality relationship using a modified Hodge star operator.

math.NA↗

arXiv2017

Yang-Mills Replacement

We develop an analog of harmonic replacement in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron. The technique, as introduced by Jost and further developed by Colding and Minicozzi, involves taking a map $v\colonΣ\to M$ defined on a surface $Σ$ and replacing its values on a small ball $B^2\subsetΣ$ with a harmonic map $u$ that has the same values as $v$ on the boundary $\partial B^2$. The resulting map on $Σ$ has lower energy, and repeating this process on balls covering $Σ$, one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection $B$ on a bundle over a four-manifold $X$, and replace it on a small ball $B^4\subset X$ with a Yang--Mills connection $A$ that has the same restriction to the boundary $\partial B^4$ as $B$. As in the harmonic replacement results of Colding and Minicozzi, we have bounds on the difference $\lVert B-A\rVert_{L^2_1(X)}^2$ in terms of the drop in energy, and we only require that the connection $B$ have small energy on the ball, rather than small $C^0$ oscillation. Throughout, we work with connections of the lowest possible regularity $L^2_1(X)$, the natural choice for this context, and so our gauge transformations are in $L^2_2(X)$ and therefore almost but not quite continuous, leading to more delicate arguments than in higher regularity.

math.DG↗

arXiv2013

Distance in the Ellipticity Graph

The ellipticity graph of a free group $F$ was defined by I. Kapovich and M. Lustig in order to study the outer automorphism group of $F$, which acts on this graph. The graph was constructed to be analogous to the curve complex of a surface. It is a bipartite graph, whose vertices are conjugacy classes of nontrivial elements of $F$ and equivalence classes of proper free product decompositions of the form $F=A*B$. A conjugacy class is joined by an edge to a free product decomposition $A*B$ whenever the conjugacy class has a representative in either $A$ or $B$. This paper uses Stallings subgroup $X$-digraphs and Whitehead automorphisms to construct algorithms that determine when the distance between two vertices of the ellipticity graph is two, for both types of vertices.

math.GR↗