Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions

Abstract

The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.