The use and analysis of infinite elements
Until recently, problems defined in infinite domains were usually solved using truncated finite elements in a finite domain. A mesh extending to some finite boundary would be chosen, and any boundary conditions which should have been applied at infinity would, in practice, be imposed at this boundary. Thus, the truncated finite element method approximates a different problem; refining the mesh leads to convergence to the solution of this new problem. Infinite elements allow one to model the behaviour at infinity, to some extent, since the basis functions used are of decaying type and the elements are of infinite length. The aim of this study is to compare these two methods, in one and two dimensions, in particular, • to derive error bounds • to show the advantages of the infinite element method • to demonstrate the effectiveness of the error bounds. The realisation of these error bounds depends on a knowledge of the true solution of the problem under examination. However, by using a spIine interpolant (or a splinefit) to the nodal infinite element parameters, we may obtain a piecewise polynomial function which is representative of the true solution, assuming that the nodal solutions are accurate. The bound which is based on the true solution may then be approximated by computing the equivalent bound obtained using this spline function. Thus we can find error bounds of real practical significance. Numerical examples and solutions are provided to illustrate the usefulness of the practical error bounds.
QA75 Electronic computers. Computer science
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