Computational Performances of Natural Element and Finite Element Methods

This paper compares the numerical performance of two numerical methods, the finite element method and the natural element method (NEM). NEM is relatively recent and is based on functions belonging to the Voronoï cell family. Although it has been proved that this method gives smoother and more accurate solutions than the finite elements, its computational cost is also known to be higher. In this paper, we compare computational efficiency, i.e., accuracy for a given cost, of finite elements and natural elements, for both Laplace and Sibson shape functions. We also bring into the comparison a Voronoï cell-based finite difference scheme which proves to be very efficient. The error is calculated using dual formulations or analytical solutions.