  A mesh-free numerical method for solving linear elliptic PDE's using the
local kernel theory that was developed for manifold learning is proposed. In
particular, this novel approach exploits the local kernel theory which allows
one to approximate the Kolmogorov operator associated with It\^o processes on
compact Riemannian manifolds without boundary or with Neumann boundary
conditions using an integral operator. Theoretical justification for the
convergence of this numerical technique is provided under the standard
conditions for the existence of the weak solutions of the PDEs. Numerical
results on various instructive examples, ranging from PDE's defined on flat and
non-flat manifolds with known and unknown embedding functions show accurate
approximation with error on the order of the kernel bandwidth parameter.
