  The accurate and efficient evaluation of Newtonian potentials over general
2-D domains is important for the numerical solution of Poisson's equation and
volume integral equations. In this paper, we present a simple and efficient
high-order algorithm for computing the Newtonian potential over a planar domain
discretized by an unstructured mesh. The algorithm is based on the use of
Green's third identity for transforming the Newtonian potential into a
collection of layer potentials over the boundaries of the mesh elements, which
can be easily evaluated by the Helsing-Ojala method. One important component of
our algorithm is the use of high-order (up to order 20) bivariate polynomial
interpolation in the monomial basis, for which we provide extensive
justification. The performance of our algorithm is illustrated through several
numerical experiments.
