This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.

翻译：本文提出了一种用于计算奇异体积积分算子的高阶精确数值方法，重点关注与二维泊松方程和亥姆霍兹方程相关的算子。借鉴边界积分算子的密度插值法思想，本方法利用格林第三恒等式以及密度函数的局部多项式插值，将体积势重新表述为单层与双层势之和，以及一个具有正则化（有界或更光滑）被积函数的体积积分。通过现有方法（如密度插值法），层势可在平面内任意位置被精确高效地计算；而正则化后的体积积分则可通过应用基本求积规则进行精确计算。相较于直接为每个奇异及近奇异体积目标点计算修正项，该方法通过将所有奇异及近奇异修正推至域边界小邻域内目标点的近奇异层势计算，显著减少了所需的专用求积计算量。文中给出了正则化与求积近似的误差估计。该方法与成熟的快速算法兼容，不仅在在线计算阶段高效，在预处理阶段同样高效。数值算例验证了所提方法的高阶精确性与效率；文中还展示了其在非均匀散射问题中的应用。