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.

翻译：在一般二维域上精确高效地评估牛顿势对于泊松方程和体积积分方程的数值求解至关重要。本文提出一种简单高效的高阶算法，用于计算由非结构化网格离散化的平面域上的牛顿势。该算法基于格林第三恒等式，将牛顿势转化为网格单元边界上的层势集合，并可通过Helsing-Ojala方法轻松评估。算法的一个关键组成部分是采用高达20阶的单项式基二元多项式插值，对此我们提供了充分的理论依据。通过多个数值实验验证了算法的性能。