Search results for "Differential form"

Probing Quantum Frustrated Systems via Factorization of the Ground State

2009

The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling transitions determines the range of applicability of mean-field descriptions in biological and physica…

Invariants of equivariant algebraic vector bundles and inequalities for dominant weights

1998

Mappings of finite distortion: Monotonicity and continuity

2001

We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev space W loc (Ω,R n), where Ω is a connected, open subset of Rn with n ≥ 2. Thus, for almost every x ∈ Ω, we can speak of the linear transformation D f(x) : Rn → Rn, called differential of f at x. Its norm is defined by |D f(x)| = sup{|D f(x)h| : h ∈ Sn−1}. We shall often identify D f(x) with its matrix, and denote by J(x, f ) = det D f(x) the Jacobian determinant. Thus, using the language of differential forms, we can write

Systematic derivation of partial differential equations for second order boundary value problems

2022

Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre-given list. This expands significantly end users possibilities to employ boundary value problems in modeling. To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems …

New degrees of freedom for differential forms on cubical meshes

2022

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Significantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.

Discrete exterior calculus for photonic crystal waveguides

2022

The discrete exterior calculus (DEC) is very promising, though not yet widely used, discretization method for photonic crystal (PC) waveguides. It can be seen as a generalization of the finite difference time domain (FDTD) method. The DEC enables efficient time evolution by construction and fits well for nonhomogeneous computational domains and obstacles of curved surfaces. These properties are typically present in applications of PC waveguides that are constructed as periodic structures of inhomogeneities in a computational domain. We present a two-dimensional DEC discretization for PC waveguides and demonstrate it with a selection of numerical experiments typical in the application area. …

Whitney forms and their extensions

2021

Whitney forms are widely known as finite elements for differential forms. Whitney’s original definition yields first order functions on simplicial complexes, and a lot of research has been devoted to extending the definition to nonsimplicial cells and higher order functions. As a result, the term Whitney forms has become somewhat ambiguous in the literature. Our aim here is to clarify the concept of Whitney forms and explicitly explain their key properties. We discuss Whitney’s initial definition with more depth than usually, giving three equivalent ways to define Whitney forms. We give a comprehensive exposition of their main properties, including the proofs. Understanding of these propert…

Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus

2022

AbstractWe present a systematic way to implement higher order Whitney forms in numerical methods based on discrete exterior calculus. Given a simplicial mesh, we first refine the mesh into smaller simplices which can be used to define higher order Whitney forms. Cochains on this refined mesh can then be interpolated using higher order Whitney forms. Hence, when the refined mesh is used with methods based on discrete exterior calculus, the solution can be expressed as a higher order Whitney form. We present algorithms for the three required steps: refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant at a given point. With our algorithms, the order of…