Issues of the Local Radial Basis Collocation Method Implementation for Solving Second Order Partial Differential Equation

Abstract

In this work, it is presented the performance of the RBF collocation method to predict the unknown field variable in non-homogeneous and variable coefficient boundary value problems. The local RBF collocation approach differs of the classical global RBF collocation approach in the way that
a radial basis function (RBF) interpolation function is defined. The former chooses to represent the meshless approximation by an expansion around a few supporting points (it constitutes a computational molecule). Any Lagragian or Hermitian RBF Hardy’s interpolation can be used to construct the meshless
locally supported shape functions which can reconstruct the field variable in each point into the molecule. In this way, several strategies have been proposed to possibly improve the imposing of the derivative boundary conditions in a strong-form approach. Three representative linear examples are solved by means different RBF collocation approaches and its results compared. It is found that all local RBF approaches performed very well. In addition, the RBF shape parameter affects the computed solution differently for each method.