The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Applications

The aim of this paper is to discuss the numerical performanceof the Galerkin method for the approximate solution of severaltwo-dimensional Fredholm integral equations of the first kindwith logarithmic kernel, and for the approximation of linearfunctionals of the solution. Predicted rates of convergenceare obtained from the theory in Sloan & Spence (1987), andthese are compared with the numerical rates for the case ofpiecewise constant approximation over equal subintervals. Thephenomenon of ‘superconvergence’ is analysed indetail and some examples are given which attain remarkably highrates of convergence.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.     

Projection Methods for Integral Equations on the Half-Line

Convergence results are proved for projection methods for integralequations of the form are such that Wiener-Hopf integral equations are included in our analysis. The convergenceresults indicate that the iterated-projection solution may exhibitsuperconvergence. The case of collocation using piecewise-constantbasis functions applied to an integral equation with kernel is discussed in detail, and numerical results are given. For this example superconvergence of the iteratedsolution, and hence also of the collocation solution at thecollocation points, is both proved theoretically and observednumerically.     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

Riccati Transformations for Eigenvalues of Systems of Linear Ordinary Differential Equations with Separated Boundary Conditions

An initial value method is presented for the calculation ofcharacteristic lengths of systems of ordinary differential equationswith separated boundary conditions. Numerical examples are includedwhich demonstrate the efficacy of the method.     

Marie Anne Paulze Lavoisier: The Mother of Modern Chemistry

Marie Anne Pierrette Paulze was a significant contributor to the understanding of chemistry in the late 1700s. Marie Anne married Antoine Laurent Lavoisier, known as the Father of Modern Chemistry, and was his chief collaborator and laboratory assistant. Marie Anne Lavoisier translated Richard Kirwans Essay on Phlogiston from English to French which allowed her husband and others to dispute Kirwans ideas. She drew many sketches and carved engravings of the laboratory instruments used by Lavoisier and his colleagues. She edited and published Lavoisiers Memoirs and hosted many parties where eminent scientists discussed new chemistry and ideas. As a result of her close work with Antoine Lavoisier, it is difficult to separate her individual contributions from his, but it is correctly assumed that much of the work accredited to him bears her fingerprints.     

Post-processing with computable error bounds for the finite element approximation of a nonlinear heat conduction problem

Email: ain{at}mcs.le.ac.uk Email: D.Kelly{at}unsw.edu.au^* Email: I.Sloan{at}unsw.edu.au^** Email: swang{at}cs.curtin.edu.au It is shown how the finite element approximation of a nonlinearheat conduction problem may be post-processed to yield enhancedapproximations to the solution and the flux at any point inthe domain. Sharp computable bounds on the accuracy of the post-processedapproximations are derived. A criterion is identified for guidingadaptive refinements of the finite element discretization. Anumerical example is given illustrating the theoretical results.     

A new pseudospectral method with upwind features

A new pseudospectral method is presented for numerical solutionsof singular perturbation problems without turning points. Boththeoretical and numerical analyses show that this new methodis an upwind scheme. It is shown that when the perturbationparameter is fixed the computed solution converges spectrallyto the exact solution as the number of collocation points tendsto infinity. ^* Supported by a Royal Fellowship Award from the Royal Societyof London.     

Invariant Imbedding, Riccati Transformations and Eigenvalue Problems

Eigenvalue calculations for systems of ordinary differentialequations using invariant imbeding and Riccati transformationstechniques have appeared independently in recent literature.The parallel development of these investigations is discussedand their equivalence demonstrated in the context of Couetteflow between rotating cylinders.     

Iterated Galerkin versus Iterated Collocation for Integral Equations of the Second Kind

We consider the numerical solution of one-dimensional Fredholmintegral equations of the second kind by the Galerkin and collocationmethods and their iterated variants, using spline bases. Inparticular, we state and prove new superconvergence resultsfor the iterated solutions, under general smoothness requirementson the kernel and solution. We find that the smoothness requirementsfor the iterated collocation method are more stringent thanthose for the iterated Galerkin method, and show by examplethat these more stringent smoothness conditions are in a certainsense necessary. In the light of these results the Galerkinand collocation schernes are compared.