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

Engaging with the research methods curriculum

Training in research methods has always been an important part of postgraduate courses; however, in recent years, what constitutes an "appropriate" kind of training for postgraduate students in Education has been shaped by national policy in addition to disciplinary traditions. Such debates became a live issue during the process of developing an online research methods module for three related MA programmes. In this paper, a critique is developed of approaches to teaching research methods. This is achieved by exploring three different approaches to the teaching and assessment of an online research methods module. The differences between these are examined, drawing on the theoretical framework and the idea of the 'engaged curriculum' developed by Barnett & Coate (2005). The paper concludes by contrasting the diversity in this case with the position currently being advocated by the UK's funding councils

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