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：本文提出一种用于评估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。遵循边界积分算子密度插值法的思路，该方法利用格林第三恒等式及密度函数的局部多项式插值，将体势重构为单层势、双层势与正则化（有界或更光滑）被积函数体积积分的组合。层势可通过现有方法（如密度插值法）在整个平面内实现精确高效评估，而正则化体积积分则可通过基本求积规则精确计算。与直接对每个奇异和近奇异体目标计算修正项相比，该方法通过将所有奇异和近奇异修正项转移至边界邻域内目标点的近奇异层势评估，显著减少了所需专用求积数量。本文提供了正则化与求积近似的误差估计。该方法与成熟快速算法兼容，不仅在在线阶段高效，且预处理阶段同样高效。数值算例证明了该高阶精度与效率特性，并展示了在非均匀散射问题中的应用。