Generalisations of the finite-element and R-matrix methods

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Abstract
A generalisation of the finite-element method is used to construct a series of highly efficient finite-basis-set methods for the solution of a one-dimensional Schrodinger equation in the continuum part of the eigenvalue spectrum. These methods are far more accurate than the standard R-matrix method and effectively eliminate the need for the correction procedures commonly applied to this earlier method. The poor convergence of the R-matrix method is found to be the result of an analogue of the Gibb's phenomenon of Fourier series expansions arising due to the inability of the basis functions to resolve the boundary conditions. It is shown how this problem can be removed by adding a few local basis functions which then allow the boundary conditions to be resolved at any energy.