A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Small- and waiting-time behaviour of the thin-film-equation

We consider the small-time behavior of interfaces of zero contact angle solutions to the thin-film equation. For a certain class of initial data, through asymptotic analyses, we deduce a wide variety of behavior for the free boundary point. These are supported by extensive numerical simulations. © 2007 Society for Industrial and Applied Mathematics

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Liberation or limitation? Understanding Iyengar Yoga as a practice of the self

This article explores the Foucauldian notions of practices of the self and care of the self, read via Deleuze, in the context of Iyengar yoga (one of the most popular forms of yoga currently). Using ethnographic and interview research data the article outlines the Iyengar yoga techniques which enable a focus upon the self to be developed, and the resources offered by the practice for the creation of ways of knowing, experiencing and forming the self. In particular, the article asks whether Iyengar yoga offers possibilities for freedom and liberation, or whether it is just another practice of control and management. Assessing Iyengar yoga via a ‘critical function’, a function of ‘struggle’ and a ‘curative and therapeutic function’, the article analyses whether the practice might constitute a mode of care of the self, and what it might offer in the context of the contemporary need to live better, as well as longer.

Becoming skilled: The cultural and corporeal geographies of teaching and learning Thai Yoga massage

This paper explores the process of learning an embodied knowledge using the work of Dreyfus and Deleuze. Although geographers have begun to acknowledge the role of embodied knowledges in social life, there have been few in-depth case studies of how these skills are learned. This paper offers a case study of Thai Yoga massage (TYM), a ‘complementary and alternative therapy’ which is growing in popularity in the United Kingdom. Having outlined the case study, the paper explores the cultural geographies of the formalisation, documentation and contestation of the set of techniques that have come to cohere in the UK as TYM. The paper then interrogates the messy corporeal geographies of learning a skill, and briefly considers how more advanced practitioners experience their skilled practice.

Post-phenomenology/post-phenomenological geographies

Post-phenomenological geographies are an emergent (and as yet relatively fragmentary) body of work. This work does not reflect a turn away from phenomenological theories; rather it reflects a critical engagement which rereads them through the post-structuralist theories of such authors as Deleuze, Derrida, and Levinas. This rereading, combined with a disciplinary context of a turn to practice and the ‘more than human’, has resulted in post-phenomenological geographies which extend the boundaries of the phenomenological focus upon the experiencing subject (in place). Thus, the interest is in the ways in which inhuman, nonhuman, and more-than-human forces contribute to processes of subject formation, place making, and inhabiting the world. These geographies have thus far been played out through critical explorations of the realms of the experiencing subject and landscape. This more-than-human focus has tested conventional human geographical methods, requiring innovative use of technologies such as video to document research, the use of experiential research methods, and also experimentation with the form of narrating these experiential methods.

A Cu tolerant population of the earthworm Dendrodrilus rubidus (Savigny, 1862) at Coniston Copper Mines, Cumbria, UK

Dendrodrilus rubidus were sampled from a mine spoil soil at Coniston Copper Mine, an abandoned Cu mine in Cumbria, UK and a Cu-free control site. Earthworms were maintained for 14 d in both Kettering loam and a Moorland soil amended with Cu nitrate. Mortality, condition index, weight change and tissue concentration were determined. In both soils D. rubidus native to the mine site were able to tolerate significantly higher soil Cu concentrations (MWRT, p <= 0.001), and exhibited significantly less change in weight (t-test, p <= 0.001) and a lower loss in condition (t-test, p <= 0.001) than control earthworms. For a given soil Cu concentration tissue Cu concentrations were greater in the mine site earthworms. Low cocoon production and viability from the mine site population prevented the determination of toxicity parameters on the F1 generation and may be an indicator of the cost of tolerance to the population. (c) 2007 Elsevier Ltd. All rights reserved.