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The performance of Kansa's method in the solution of elliptic PDEswith singular boundary conditions is addressed. Like in all global numerical schemes, low-order singularities bring about Gibbs' oscillations that deteriorate the accuracy and the convergence rate of Kansa's method to a great extent. Moreover, they may render it uncapable of handling common problems of incompressible flow. Focussing on a problem of Laplacian flow which is linked to the benchmark Motz problem, it is shown how all these dfficulties can be overcome by enriching the RBF interpolant with the proper singularity-capturing terms. This simple modification even enables Kansa's method to outperform FEM in the conservation of Laplacianflow through an irregular channel.