A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built-in multiresolution features in a very attractive meshfree method for numerical solution of partial dif ferential equations. The design and implementation of a Galerkin-based repr oducing kernel particle method, however, faces several challenges such as t he issue of nodal volumes and accurate and efficient implementation of boun dary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary co nditions are not critical issues and points can be sprinkled randomly makin g the point collocation method a true meshless approach. The point collocat ion method based on reproducing kernel approximations, however, requires th e calculation of higher-order derivatives that would typically not be requi red in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The po int collocation method is shown to be accurate for several one and two-dime nsional problems and the convergence rate of the point collocation method i s addressed. Copyright (C) 2000 John Wiley & Sons, Ltd.