Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Pushing the boundaries of the social: private agri-food standards and the governance of Fair Trade in European public procurement

The article explores how fair trade and associated private agri-food standards are incorporated into public procurement in Europe. Procurement law is underpinned by principles of equity, non-discrimination and transparency; one consequence is that legal obstacles exist to fair trade being privileged within procurement practice. These obstacles have pragmatic dimensions, concerning whether and how procurement can be used to fulfil wider social policy objectives or to incorporate private standards; they also bring to the fore underlying issues of value. Taking an agency-based approach and incorporating the concept of governability, empirical evidence demonstrates the role played by different actors in negotiating fair trade’s passage into procurement through pre-empting and managing legal risk. This process exposes contestations that arise when contrasting values come together within sustainable procurement. This examination of fair trade in public procurement helps reveal how practices and knowledge on ethical consumption enter into a new governance arena within the global agri-food system.

Proceedings of the inaugural construction management and economics ‘Past, Present and Future’ conference CME25, 16-18 July 2007, University of Reading, UK

This conference was an unusual and interesting event. Celebrating 25 years of Construction Management and Economics provides us with an opportunity to reflect on the research that has been reported over the years, to consider where we are now, and to think about the future of academic research in this area. Hence the sub-title of this conference: “past, present and future”. Looking through these papers, some things are clear. First, the range of topics considered interesting has expanded hugely since the journal was first published. Second, the research methods are also more diverse. Third, the involvement of wider groups of stakeholder is evident. There is a danger that this might lead to dilution of the field. But my instinct has always been to argue against the notion that Construction Management and Economics represents a discipline, as such. Granted, there are plenty of university departments around the world that would justify the idea of a discipline. But the vast majority of academic departments who contribute to the life of this journal carry different names to this. Indeed, the range and breadth of methodological approaches to the research reported in Construction Management and Economics indicates that there are several different academic disciplines being brought to bear on the construction sector. Some papers are based on economics, some on psychology and others on operational research, sociology, law, statistics, information technology, and so on. This is why I maintain that construction management is not an academic discipline, but a field of study to which a range of academic disciplines are applied. This may be why it is so interesting to be involved in this journal. The problems to which the papers are applied develop and grow. But the broad topics of the earliest papers in the journal are still relevant today. What has changed a lot is our interpretation of the problems that confront the construction sector all over the world, and the methodological approaches to resolving them. There is a constant difficulty in dealing with topics as inherently practical as these. While the demands of the academic world are driven by the need for the rigorous application of sound methods, the demands of the practical world are quite different. It can be difficult to meet the needs of both sets of stakeholders at the same time. However, increasing numbers of postgraduate courses in our area result in larger numbers of practitioners with a deeper appreciation of what research is all about, and how to interpret and apply the lessons from research. It also seems that there are contributions coming not just from construction-related university departments, but also from departments with identifiable methodological traditions of their own. I like to think that our authors can publish in journals beyond the construction-related areas, to disseminate their theoretical insights into other disciplines, and to contribute to the strength of this journal by citing our articles in more mono-disciplinary journals. This would contribute to the future of the journal in a very strong and developmental way. The greatest danger we face is in excessive self-citation, i.e. referring only to sources within the CM&E literature or, worse, referring only to other articles in the same journal. The only way to ensure a strong and influential position for journals and university departments like ours is to be sure that our work is informing other academic disciplines. This is what I would see as the future, our logical next step. If, as a community of researchers, we are not producing papers that challenge and inform the fundamentals of research methods and analytical processes, then no matter how practically relevant our output is to the industry, it will remain derivative and secondary, based on the methodological insights of others. The balancing act between methodological rigour and practical relevance is a difficult one, but not, of course, a balance that has to be struck in every single paper.

Evaluation of a boundary integral representation for the conformal mapping of the unit disk onto a simply-connected domain

A representation of the conformal mapping g of the interior or exterior of the unit circle onto a simply-connected domain Ω as a boundary integral in terms ofƒ|∂Ω is obtained, whereƒ :=g -l. A product integration scheme for the approximation of the boundary integral is described and analysed. An ill-conditioning problem related to the domain geometry is discussed. Numerical examples confirm the conclusions of this discussion and support the analysis of the quadrature scheme.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Standing on the shoulders of giants: an examination of the interdisciplinary foundation of relationship marketing

In the 20th century, the scholarly study of human relationships both grew dramatically and simultaneously fragmented into various disciplines and subdisciplines. Although diversity of thought is generally considered helpful for the evolution of scientific fields, the value accrued from interdisciplinary discourse depends on the ability of scholars to integrate multiple perspectives and synthesize foundational works in a systematic manner. The goal of this study is to synthesize foundational theories from social and behavioral sciences that have contributed to an understanding of relationship marketing. In seeking to provide a holistic understanding of the field, we incorporate contributions from the disciplines of marketing, management, psychology, and sociology. In building on our analysis, we synthesize our findings into a conceptual model that examines the systematic dimensions of relationship marketing. The article concludes by identifying key themes for contributors to the Journal of Relationship Marketing to consider going forward.

Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms u(tt) - Delta u - gamma(t)Delta u(t) + beta(epsilon)(t)u(t) = f(u), in a bounded domain Omega subset of R(n), under some restrictions on beta(epsilon)(t), gamma(t) and growth restrictions on the nonlinear term f. The function beta(epsilon)(t) depends on a parameter epsilon, beta(epsilon)(t) -> 0. We will prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {A(epsilon)(t) : t is an element of R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at epsilon = 0. (C) 2010 Elsevier Ltd. All rights reserved.

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods. the sub and supersolutions method, comparison principles and a priori estimates. we study existence, multiplicity, and the behavior with respect to lambda of positive solutions of p-Laplace equations of the form -Delta(p)u = lambda h(x, u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x, a(x)) = 0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros. (C) 2009 Elsevier Inc. All rights reserved.

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Omega(0) which is interior to the physical domain Omega subset of R(n). We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Omega(0) and converges uniformly to a continuous and positive function in Omega(1) = (Omega) over bar\Omega(0). (C) 2009 Elsevier Inc. All rights reserved.

An extension of the concept of gradient semigroups which is stable under perturbation

In this article we introduce the concept of a gradient-like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup (those possessing a Lyapunov function, see [J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1989]) and a nonlinear semigroup possessing a gradient-like attractor. We prove that a perturbation of a gradient-like nonlinear semigroup remains a gradient-like nonlinear semigroup. Moreover, for non-autonomous dynamical systems we introduce the concept of a gradient-like evolution process and prove that a non-autonomous perturbation of a gradient-like nonlinear semigroup is a gradient-like evolution process. For gradient-like nonlinear semigroups and evolution processes, we prove continuity, characterization and (pullback and forwards) exponential attraction of their attractors under perturbation extending the results of [A.N. Carvalho, J.A. Langa, J.C. Robinson, A. Suarez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007) 570-603] on characterization and of [A.V. Babin, M.I. Vishik, Attractors in Evolutionary Equations, Stud. Math. Appl.. vol. 25, North-Holland, Amsterdam, 1992] on exponential attraction. (C) 2009 Elsevier Inc. All rights reserved.

On the Fucik spectrum of the Laplacian on a torus

We study the Fucik spectrum of the Laplacian on a two-dimensional torus T(2). Exploiting the invariance properties of the domain T(2) with respect to translations we obtain a good description of large parts of the spectrum. In particular, for each eigenvalue of the Laplacian we will find an explicit global curve in the Fucik spectrum which passes through this eigenvalue; these curves are ordered, and we will show that their asymptotic limits are positive. On the other hand, using a topological index based on the mentioned group invariance, we will obtain a variational characterization of global curves in the Fucik spectrum; also these curves emanate from the eigenvalues of the Laplacian, and we will show that they tend asymptotically to zero. Thus, we infer that the variational and the explicit curves cannot coincide globally, and that in fact many curve crossings must occur. We will give a bifurcation result which partially explains these phenomena. (C) 2008 Elsevier Inc. All rights reserved.