11 citations · 50 across the 13 of their papers we have counts for
15 papers
Finite Element Spaces of Double Two-Forms With Polynomial Coefficients
Yakov Berchenko-Kogan, Lily DiPaulo
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 matri…
Finite element spaces of double forms
Yakov Berchenko-Kogan, Evan S. Gawlik
The tensor product of two differential forms of degree and is a multilinear form that is alternating in its first arguments and alternating in its last arguments. T…
Extension operators and geometric decompositions
Yakov Berchenko-Kogan
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,…
Blow-up Whitney forms, shadow forms, and Poisson processes
Yakov Berchenko-Kogan, Evan S. Gawlik
The Whitney forms on a simplex admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Bras…
Symmetric bases for finite element exterior calculus spaces
Yakov Berchenko-Kogan
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…
Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
Yakov Berchenko-Kogan, Evan S. Gawlik
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 R…
Numerically computing the index of mean curvature flow self-shrinkers
Yakov Berchenko-Kogan
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. Singulari…
Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori
Yakov Berchenko-Kogan
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 dilation…
Charge-conserving hybrid methods for the Yang-Mills equations
Yakov Berchenko-Kogan, Ari Stern
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. Christi…
Constraint-preserving hybrid finite element methods for Maxwell's equations
Yakov Berchenko-Kogan, Ari Stern
Maxwell's equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints cor…
Duality in finite element exterior calculus and Hodge duality on the sphere
Yakov Berchenko-Kogan
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…
The entropy of the Angenent torus is approximately 1.85122
Yakov Berchenko-Kogan
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 example…
Duality in finite element exterior calculus
Yakov Berchenko-Kogan
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 ,…
Yang-Mills Replacement
Yakov Berchenko-Kogan
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 Jos…
Distance in the Ellipticity Graph
Yakov Berchenko-Kogan
The ellipticity graph of a free group was defined by I. Kapovich and M. Lustig in order to study the outer automorphism group of , which acts on this graph. The graph was co…