On the Numerical Solution of Some Elliptic PDEs with Neumann Boundary Conditions through Multinode Shepard Method

Abstract

In this talk, the multinode Shepard method is proposed to solve elliptic partial differential equations with Neumann boundary conditions. The method has been opportunely handled to solve different equations with various boundary conditions dealing with scattered distribution of points [1, 2]. The particular feature of the method, based on local polynomial interpolants on opportunely choosen nearby nodes [3], is a collocation matrix which is reduced in size with many zero entrances and a small condition number. Experiments in 2d domains have been performed with Neumann boundary conditions. Comparisons with the analytic solutions and the results generated with the RBF method proposed by Kansa are presented referring to different distribution of points.