Positivity conditions in meshless collocation methods

Abstract

Collocation meshless methods are conceptually simple, easy-to-implement and fast numerical methods. The robustness of collocation methods has, however, been an issue especially for scattered set of points. In this paper we show that the robustness of collocation meshless methods can be improved by ensuring that certain conditions, defined as the positivity conditions, are satisfied when constructing approximation functions and their derivatives. The significance of positivity conditions is pointed out by an error analysis of the finite cloud method, which is a collocation based meshless method. We propose techniques, based on modification of weighting functions, to ensure satisfaction of positivity conditions on the approximation function and its derivatives when using a scattered set of points. Several types of weighting functions are tested for 1D and 2D problems on scattered points. Numerical results demonstrate the effectiveness of collocation methods when positivity conditions are satisfied.