Langrangian finite element and finite difference methods for poisson problems

The use of Lagrangian finite element methods for solving a Poisson problem produces systems of linear equations, the global stiffness equations. The components of the vectors which are the solutions of these systems are approximations to the exact solution of the problem at nodal points in the region of definition. There is thus associated with each nodal point an equation which can be thought of as a difference equation. Difference equations resulting from the use of polynomial trial functions of various orders on regular meshes of square and isosceles right triangular elements are derived. The rival merits of this technique of setting up a standard difference equation, as distinct from the more usual practice with finite elements of the repeated use of local stiffness matrices, are considered

A numerical conformal transformation method for harmonic mixed boundary value problems in polygonal domains

A method is given for solving two dimensional harmonic mixed boundary value problems in simply-connected polygonal domains with re-entrant boundaries. The method consists of a numerical conformal mapping together with three other conformal transformations. The numerical mapping transforms the original domain onto the unit circle, which in turn is mapped onto a rectangle by means of two bilinear and one Schwarz-Christoffel transformations. The transformed problem in the rectangle is solved by inspection

Numerical solution of two dimensional harmonic boundary problems containing singularities by conformal transformation methods

Numerical solutions to a class of two dimensional harmonic mixed boundary value problems defined on rectangular domains and containing singularities are obtained using conformal transformation methods. These map the original problems into similar ones containing no singularities, and to which analytic solutions are known. Although the mapping technique produces analytic solutions to the original problems, these involve elliptic functions and integrals which have to be evaluated numerically, so that in practice only approximations can be obtained. Results calculated in this manner for model problems compare favourably with those obtained previously by other methods. On this evidence, and because of the ease with which the method can be adapted to different individual problems, we strongly recommend the transformation technique for solving problems of this class. W

Cubic spline interpolation of harmonic functions

It is shown that for the two dimensional Laplace equation a univariate cubic spline approximation in either space direction together with a difference approximation in the other leads to the well-known nine-point finite-difference formula. For harmonic problems defined in rectangular regions this property provides a means of determining with ease accurate approximations at any point in the region

A cubic spline technique for the one dimensional heat conduction equation

A new method is developed for the numerical solution of the heat conduction equation in one space dimension by replacing the space derivative with a cubic spline approximation and the time derivative with a finite- difference approximation. The method is equivalent to a new finite-difference scheme and produces at each time level an interpolating spline function

Diagramming the Social: Relational Method in Research

This book challenges the hyper-production and proliferation of concepts in modern social research. It presents a distinctive methodological response to this tendency through an exploration of one of the most underappreciated yet widely deployed conventions for the analysis of social processes: the creation of diagrammatic relational spaces. Designed to capture social processes in a way that resists reductive and essentialist categories, such spaces have the capacity to produce powerful, systematic analyses that break the spell of concept proliferation and its resultant naively realist approach to explaining the world. Through an exploration of key examples and series of original case studies, the authors demonstrate the application of this approach across a variety of empirical settings and academic disciplines. They thus offer a relational and pragmatic approach to social research that resists current trends characterised by supposedly self-evident data and/or disconnected theory. As such, the book constitutes an important contribution to some of the central questions in current social research, and promises to unsettle and reinvigorate considerations of method across different fields of practice

Giant Relaxation Oscillations in a Very Strongly Hysteretic SQUID ring-Tank Circuit System

In this paper we show that the radio frequency (rf) dynamical characteristics of a very strongly hysteretic SQUID ring, coupled to an rf tank circuit resonator, display relaxation oscillations. We demonstrate that the the overall form of these characteristics, together with the relaxation oscillations, can be modelled accurately by solving the quasi-classical non-linear equations of motion for the system. We suggest that in these very strongly hysteretic regimes SQUID ring-resonator systems may find application in novel logic and memory devices.Comment: 7 pages, 5 figures. Uploaded as implementing a policy of arXiving old paper

Pinch Resonances in a Radio Frequency Driven SQUID Ring-Resonator System

In this paper we present experimental data on the frequency domain response of a SQUID ring (a Josephson weak link enclosed by a thick superconducting ring) coupled to a radio frequency (rf) tank circuit resonator. We show that with the ring weakly hysteretic the resonance lineshape of this coupled system can display opposed fold bifurcations that appear to touch (pinch off). We demonstrate that for appropriate circuit parameters these pinch off lineshapes exist as solutions of the non-linear equations of motion for the system.Comment: 9 pages, 8 figures, Uploaded as implementing a policy of arXiving old paper