  This paper presents a new approach for solving the close evaluation problem
in three dimensions, commonly encountered while solving linear elliptic partial
differential equations via potential theory. The goal is to evaluate layer
potentials close to the boundary over which they are defined. The approach
introduced here converts these nearly-singular integrals on a patch of the
boundary to a set of non-singular line integrals on the patch boundary using
the Stokes theorem on manifolds. A function approximation scheme based on
harmonic polynomials is designed to express the integrand in a form that is
suitable for applying the Stokes theorem. As long as the data -- the boundary
and the density function -- is given in a high-order format, the double-layer
potential and its derivatives can be evaluated with high-order accuracy using
this scheme both on and off the boundary. In particular, we present numerical
results demonstrating seventh-order convergence on a smooth, warped torus
example achieving 10-digit accuracy in evaluating double layer potential at
targets that are arbitrarily close to the boundary.
