Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent both and its corresponding local limit as interaction vanishes. analysis carried out on a special family rectilinear Cartesian grids linear designed support. key idea the stability compare standard Galerkin which stable. In addition, there large computational cost assembling stiffness matrix problem because high order Gaussian quadrature usually needed evaluate integral. thus provide remedy by introducing quasi-discrete operator no numerical further after applying scheme. combined correct taking limits spatial resolution simultaneously. theoretical results then validated experiments. additionally illustrate connection between proposed technique existing optimization based approach generalized moving least squares (GMLS).