Effect of Local Support Configuration on the Precision of Numerical Solutions of Poisson Equation Obtained With Differential Quadrature Method

Differential quadrature methods are devised to numerically solve ordinary and partial differential equations by approximating the derivatives of the unknown function at points of a cloud defined on the domain of interest as weighted sums of the values of such function at other points of the cloud. Local versions of this class of meshless methods restrict the points used in such expansion, by establishing suitable supporting regions. In this paper, we present the local differential quadrature method and we use it to solve a boundary problem in electromagnetism. In order to do this, we evaluate the numerical solutions of the Poisson equation on a 2-D domain. Furthermore, we propose an alternative definition of supporting region that has yielded better solutions than the conventional one. Root-mean-square errors for the approximations with both (alternative and conventional) definitions of local supports are obtained and their dependences with the density of nodes are studied. We find out that the best accuracy obtained with the alternative definition of the local support is due to the smaller condition numbers of the linear systems yielded.