On the enriched RBF method for singular potential problems

The performance of Kansaʹs method in the solution of elliptic partial differential equations (PDEs) with 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 Kansaʹs 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 difficulties can be overcome by enriching the radial basis function (RBF) interpolant with the proper singularity-capturing terms. This simple modification even enables Kansaʹs method to outperform the finite element method (FEM) in the conservation of Laplacian flow through an irregular channel.