Greg Fasshauer

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In this study the authors present an application of the radial basis function-pseudospectral {(RBF-PS)} meshfree method as well as a least squares variant thereof to a three-dimensional {(3D)} benchmark engineering problem defined by the Laplace equation. To their knowledge this is the first such study. The {RBF-PS} method is a version of the radial basis function {(RBF)} collocation method formulated in the vein of traditional pseudospectral methods. The least squares {RBF-PS} method introduced here as a modification of the {RBF-PS} method allows the authors to work with fewer {RBFs} while maintaining a high number of collocation points. In addition, the authors use a leave-one-out cross validation algorithm to choose an `optimal` shape parameter for their basis functions. In order to evaluate the accuracy, effectiveness and applicability of their new approach, the authors apply it to a {3D} benchmark electromagnetic problem. Their numerical results demonstrate that the proposed methods compare favourably to the finite difference and finite element methods.