arXiv subjects

Douglas N. Arnold

Publications and source records attributed to Douglas N. Arnold.

At least 19 recordsLinked to original sources

arXiv2021

Sharp estimates for the integrated density of states in Anderson tight-binding models

Recent work [G. David, M. Filoche, and S. Mayboroda, arXiv:1909.10558[Adv. Math. (to be published)]] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schrödinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.

math-ph↗

arXiv2021

The landscape law for tight binding Hamiltonians

The present paper extends the landscape theory pioneered in [FM, ADFJM2, DFM] to the tight-binding Schrödinger operator on $\Z^d$. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape.

math-ph↗

arXiv2021

Local $L^2$-bounded commuting projections in FEEC

We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in $L^2$ and commute with the exterior derivative.

math.NA↗

arXiv2021

Complexes from complexes

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincaré type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

math.NA↗

arXiv2020

The Hellan-Herrmann-Johnson method with curved elements

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babuška paradox, highlighting the geometrically non-conforming aspect of the HHJ method.

math.NA↗

arXiv2018

Localization of eigenfunctions via an effective potential

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $Ω$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to $Lu=1$, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, $1/u$, acts as an effective potential. Hence from the single measurement of $u$, we obtain, via $1/u$, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum. (This version strengthens and simplifies the results of the first one by replacing a global bi-Lipschitz hypothesis on the domain with a local bi-Lipschitz hypothesis. It improves on the second version by adding pictures and numerical examples. This version is identical to the third version; all that is changed is to correct some tex mistakes in symbols in this abstract. There are no changes to the paper itself.)

math.AP↗

arXiv2016

Finite element exterior calculus with lower-order terms

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that surprisingly for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.

math.NA↗

arXiv2016

Effective confining potential of quantum states in disordered media

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

cond-mat.dis-nn↗

arXiv2014

Finite element differential forms on curvilinear cubic meshes and their approximation properties

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.

math.NA↗

arXiv2014

Mixed finite elements for elasticity on quadrilateral meshes

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer $r \geq 2$ and with rate of convergence in the $L^2$ norm of order $r$ for all the variables. The methods use Raviart-Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.

math.NA↗

arXiv2014

On the consistency of the combinatorial codifferential

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to 1-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend Smits's result to arbitrary dimensions, showing that the combinatorial codifferential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codifferential is not consistent even for the most regular subdivision process.

math.DG↗

arXiv2013

Mixed methods for elastodynamics with weak symmetry

We analyze the application to elastodynamic problems of mixed finite element methods for elasticity with weak symmetry. Our approach leads to a semidiscrete method which consists of a system of ordinary differential equations without algebraic constraints. Our error analysis, which is based on a new elliptic projection operator, applies to several mixed finite element spaces developed for elastostatics. The error estimates we obtain are robust for nearly incompressible materials.

math.NA↗

arXiv2013

Finite element differential forms on cubical meshes

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.

math.NA↗

arXiv2013

Nonconforming tetrahedral mixed finite elements for elasticity

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.

math.NA↗

arXiv2012

Spaces of finite element differential forms

We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit commuting projections. We present two families of spaces in the case of simplicial meshes, and two other families in the case of cubical meshes. We make use of the exterior calculus and the Koszul complex to define and understand the spaces. These tools allow us to treat a wide variety of situations, which are often treated separately, in a unified fashion.

math.NA↗

arXiv2012

Finite element exterior calculus for parabolic problems

In this paper, we consider the extension of the finite element exterior calculus from elliptic problems, in which the Hodge Laplacian is an appropriate model problem, to parabolic problems, for which we take the Hodge heat equation as our model problem. The numerical method we study is a Galerkin method based on a mixed variational formulation and using as subspaces the same spaces of finite element differential forms which are used for elliptic problems. We analyze both the semidiscrete and a fully-discrete numerical scheme.

math.NA↗

arXiv2012

Mathematicians take a stand

We survey the reasons for the ongoing boycott of the publisher Elsevier. We examine Elsevier's pricing and bundling policies, restrictions on dissemination by authors, and lapses in ethics and peer review, and we conclude with thoughts about the future of mathematical publishing.

math.HO↗

arXiv2011

Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

math.NA↗