Corrected collocation approximations for the harmonic dirichlet problem

Collocation approximations with harmonic basis functions to the solution of the harmonic Dirichlet problem are investigated. The choice of collocation points for a best local approximation is discussed, and a result is given in terms of the abscissae of some best quadrature formulae. A global near-best approximation is obtained by adding a correction term to the collocation approximation, utilizing basic properties of the Green's function. Numerical examples are given, demonstrating the great improvement achieved. The same correction term can also improve on least-squares approximations and Galerkin approximations, and the results can easily be adapted to deal with mixed harmonic boundary value problems.