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A computational scheme for solving 2D Laplace boundary-value problems usingrational functions as the basis functions is described. The scheme belongs tothe class of desingularized methods, for which the location of singularitiesand testing points is a major issue that is addressed by the proposed scheme,in the context of the 2D Laplace equation. Well-established rational-functionfitting techniques are used to set the poles, while residues are determined byenforcing the boundary conditions in the least-squares sense at the nodes ofrational Gauss-Chebyshev quadrature rules. Numerical results show that errorsapproaching the machine epsilon can be obtained for sharp and almost sharpcorners, nearly-touching boundaries, and almost-singular boundary data. We showvarious examples of these cases in which the method yields compact solutions,requiring fewer basis functions than the Nystr\"{o}m method, for the sameaccuracy. A scheme for solving fairly large-scale problems is also presented.

On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems