  In this work, we study the reproducing kernel (RK) collocation method for the
peridynamic Navier equation. We first apply a linear RK approximation on both
displacements and dilatation, then back-substitute dilatation, and solve the
peridynamic Navier equation in a pure displacement form. The RK collocation
scheme converges to the nonlocal limit and also to the local limit as nonlocal
interactions vanish. The stability is shown by comparing the collocation scheme
with the standard Galerkin scheme using Fourier analysis. We then apply the RK
collocation to the quasi-discrete peridynamic Navier equation and show its
convergence to the correct local limit when the ratio between the nonlocal
length scale and the discretization parameter is fixed. The analysis is carried
out on a special family of rectilinear Cartesian grids for the RK collocation
method with a designated kernel with finite support. We assume the Lam\'{e}
parameters satisfy $\lambda \geq \mu$ to avoid adding extra constraints on the
nonlocal kernel. Finally, numerical experiments are conducted to validate the
theoretical results.
