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Abstract: The performance of Kansa''s 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 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 difficulties can be overcome by enriching the radial basis function (RBF) interpolant with the proper singularity-capturing terms. This simple modification even enables Kansa''s method to outperform the finite element method (FEM) in the conservation of Laplacian flow through an irregular channel. [Copyright &y& Elsevier]

Abstract: In this paper we consider the problem of injecting a non-Newtonian fluid into a thin cavity. Using the Hele–Shaw approximation the problem reduces to a moving boundary problem in which the pressure is described by a 2D, nonlinear, elliptic equation. Mesh-free methods are very well suited for the numerical solution of moving boundary problems since no remeshing is needed at each time step to correctly represent the boundary. Among these methods, we have chosen the asymmetric RBF collocation method (Kansa''s method) to compute the pressure distribution. Kansa''s method is truly mesh-free, and is one of the most frequently used methods due to its accuracy and ease of implementation. Once the pressure is known, the velocity at each point in the moving front is computed and, therefore, the location of the front can be updated. To efficiently model the front motion we use the level set method, which is very accurate, and can handle both front collisions and front break-ups without difficulty. [Copyright &y& Elsevier]