  In this paper, we extend the class of kernel methods, the so-called diffusion
maps (DM), and its local kernel variants, to approximate second-order
differential operators defined on smooth manifolds with boundaries that
naturally arise in elliptic PDE models. To achieve this goal, we introduce the
Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified
by the set of point clouds on the unknown original manifold together with a set
of ghost points, specified along the estimated tangential direction at the
sampled points at the boundary. The resulting GPDM estimator restricts the
standard DM matrix to a set of extrapolation equations that estimates the
function values at the ghost points. This adjustment is analogous to the
classical ghost point method in finite-difference scheme for solving PDEs on
flat domain. As opposed to the classical DM which diverges near the boundary,
the proposed GPDM estimator converges pointwise even near the boundary.
Applying the consistent GPDM estimator to solve the well-posed elliptic PDEs
with classical boundary conditions (Dirichlet, Neumann, and Robin), we
establish the convergence of the approximate solution under appropriate
smoothness assumptions. We numerically validate the proposed mesh-free PDE
solver on various problems defined on simple sub-manifolds embedded in
Euclidean spaces as well as on an unknown manifold. Numerically, we also found
that the GPDM is more accurate compared to DM in solving elliptic eigenvalue
problems on bounded smooth manifolds.
