Osculatory interpolation in the method of fundamental solution for nonlinear Poisson problems

The Method of Fundamental Solution (also known as the F-Trefftz method or the singularity method) is an efficient numerical method for the solution of Laplace equation for both two- and three-dimensional problems. In recent years, the method has also been applied for the solution of Poisson equations by finding the particular solution to the nonhomogeneous terms. In general, approximate particular solutions are constructed using the interpolation of the nonhomogeneous terms by the radial basis functions. The method has been validated in recent papers. This paper presents an improvement of the solution procedure for such problems. The improvement is achieved by using radial basis functions called osculatory radial basis functions. Such functions make use of the normal gradient at boundary to obtain improved interpolation. The efficacy of the method is demonstrated for some prototypical nonlinear Poisson problems and for multiple Poisson equations.