  When using the boundary integral equation method to solve a boundary value
problem, the evaluation of the solution near the boundary is challenging to
compute because the layer potentials that represent the solution are
nearly-singular integrals. To address this close evaluation problem, we apply
an asymptotic analysis of these nearly singular integrals and obtain an
asymptotic approximation. We derive the asymptotic approximation for the case
of the double-layer potential in two and three dimensions, representing the
solution of the interior Dirichlet problem for Laplace's equation. By doing so,
we obtain an asymptotic approximation given by the Dirichlet data at the
boundary point nearest to the interior evaluation point plus a nonlocal
correction. We present numerical methods to compute this asymptotic
approximation, and we demonstrate the efficiency and accuracy of the asymptotic
approximation through several examples. These examples show that the asymptotic
approximation is useful as it accurately approximates the close evaluation of
the double-layer potential while requiring only modest computational resources.
