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.Comment: v2: as accepted by Mathematics of Computation after minor revisions; v3: this version corresponds to the final version for Math. Comp., after copyediting and galley proof

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.Comment: 13 pages, 2 figure

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner Transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive 'interference terms', which often make computation impractical. In this connection, we propose the use of the smoothed Wigner Transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The 'taming' of the interference terms by the SWT is illustrated, and an asymptotic model for the Schroedinger equation is constructed and numerically verified.Comment: 16 pages, 8 figure

Development and Validation of the Geriatric Anxiety Inventory

Background: Anxiety symptoms and anxiety disorders are highly prevalent among elderly people, although infrequently the subject of systematic research in this age group. One important limitation is the lack of a widely accepted instrument to measure dimensional anxiety in both normal old people and old people with mental health problems seen in various settings. Accordingly, we developed and tested of a short scale to measure anxiety in older people. Methods:We generated a large number of potential items de novo and by reference to existing anxiety scales, and then reduced the number of items to 60 through consultation with a reference group consisting of psychologists, psychiatrists and normal elderly people. We then tested the psychometric properties of these 60 items in 452 normal old people and 46 patients attending a psychogeriatric service. We were able to reduce the number of items to 20. We chose a 1-week perspective and a dichotomous response scale. Results: Cronbach's alpha for the 20-item Geriatric Anxiety Inventory (GAI) was 0.91 among normal elderly people and 0.93 in the psychogeriatric sample. Concurrent validity with a variety of other measures was demonstrated in both the normal sample and the psychogeriatric sample. Inter-rater and test-retest reliability were found to be excellent. Receiver operating characteristic analysis indicated a cut-point of 10/11 for the detection of DSM-IV Generalized Anxiety Disorder (GAD) in the psychogeriatric sample, with 83% of patients correctly classified with a specificity of 84% and a sensitivity of 75%. Conclusions: The GAI is a new 20-item self-report or nurse-administered scale that measures dimensional anxiety in elderly people. It has sound psychometric properties. Initial clinical testing indicates that it is able to discriminate between those with and without any anxiety disorder and between those with and without DSM-IV GAD

Comparison of cluster algorithms for the bond-diluted Ising model

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times Formula Presented and Formula Presented of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size Formula Presented when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time Formula Presented to be bounded from below by Formula Presented with a dynamical exponent Formula Presented for a bond concentration Formula Presented. Furthermore, we numerically show that this lower bound is actually taken for several values of Formula Presented in the range Formula Presented. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at Formula Presented, we find that its dynamical exponent is Formula Presented

A comparison of cluster algorithms for the bond-diluted Ising model

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τw and τsw of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τw to be bounded from below by Lzw with a dynamical exponent zw=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is zsw=0.09(4)