Reproducing polynomial (singularity) particle methods and adaptive meshless methods for two-dimensional elliptic boundary value problems

Oh et al. [H.-S. Oh, J.G. Kim, J.W. Jeong, The closed form reproducing polynomial particle shape functions for meshfree particle methods, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3435–3461] introduced the reproducing polynomial particle (RPP) shape functions that are piecewise polynomial and satisfy the Kronecker delta property. In this paper, we introduce reproducing polynomial particle methods (RPPM) that is the Galerkin approximation method associated with the use of the RPP approximation space. Planting particles in the computation domain in a patchwise uniform manner, we also introduce patchwise RPPM. Furthermore, constructing partition of unity functions with flat-top adaptively, we introduce adaptive patchwise RPPM and compare it with reproducing singularity particle methods (RSPM) that is the Galerkin approximation method associated with the use of reproducing singularity particle (RSP) shape functions.