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.