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 differential equations. The design and implementation of a Galerkin-based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary 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 conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher-order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two-dimensional problems and the convergence rate of the point collocation method is addressed.