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