Higher order multipoint method – from Collatz to meshless FDM

Abstract

The higher order multipoint meshless method for boundary value problems is considered in this paper. The new method relies on the Collatz multipoint concept and the meshless FDM.The main idea of multipoint technique is based on raising the local approximation order of a searched function by assuming additional degrees of freedom at each node of stencil, e.g. by including a combination of nodal values of the right-hand side of considered differential equation. Thus, the FD formula takes into account a combination of unknown function values which is equal a combination of additional d.o.f., e.g. right-hand side of PDE. In this way one may generate higher order FD operators without any additional unknowns using the same set of nodes in stencil as in the non-multipoint case.Essential modifications and extensions of the old classical multipoint formulation have been introduced for the purpose of this research. New fully automatic multipoint meshless FDM uses the moving weighted least squares approximation instead of the interpolation proposed by Collatz, and is based on arbitrarily distributed cloud of nodes. Moreover, besides the local formulation, also various global formulations of b.v. problems are possible.Several numerical benchmark problems analyzed illustrate the effectiveness of the proposed approach.