Improved point-matching techniques

An exposition of point-matching techniques, which yield approximate solutions to two-dimensional boundary-value problems, is presented for examples in heat conduction. Improvements of the method and extensions to initial-value and three-dimensional problems are also discussed. Application of this approach to domains with irregular boundaries and internal cut-outs presents no particular difficulty, since imposed conditions are satisfied only pointwise by known exact solutions to the governing differential equations. Since conditions satisfied do not correspond completely to those imposed, the resulting equations are non-unique approximations to the exact solutions. An alternative point of view which may be associated with point-matching is that solutions so obtained are exact for 'approximate problems, ' wherein boundary and/or initial conditions are those obtained naturally through application of the method.Three techniques for improving results of this semi-collocative procedure are presented, and bounds on errors attendant to the method are obtained for a class of conduction problems. Numerical examples treated include steady-state heat flow in two-dimensional solids, and a one-dimensional transient problem. Good oagreement between point-matched and 'exact' results are obtained in all cases considered.