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：针对一般二维域上牛顿势的精确高效评估对于泊松方程和体积积分方程的数值求解至关重要。本文提出了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元素边界上一系列层势的集合，可通过Helsing-Ojala方法轻松评估。算法的关键组成部分之一是采用单变量基下的高阶（最高至20阶）双变量多项式插值，我们对此提供了充分的理论依据。通过多个数值实验展示了算法的性能。