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. We describe the method both for domains meshed by mapped quadrilaterals and triangles, introducing for each case (i) well-conditioned methods for the production of certain requisite source polynomial interpolants and (ii) efficient translation formulae for polynomial particular solutions. 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.

翻译：本文提出了一种高精度数值方法，用于评估奇异体积积分算子，重点关注与二维泊松方程和亥姆霍兹方程相关的算子。该方法借鉴边界积分算子密度插值法的思想，利用格林第三恒等式和密度函数的局部多项式插值，将体积势重新表述为单层势、双层势以及具有正则化（有界或更光滑）被积函数的体积积分之和。通过现有方法（如密度插值法）可在全平面高效精确地评估层势，而正则化体积积分则可通过基本求积法则精确计算。我们针对由映射四边形和三角形网格化的区域分别描述了该方法，并为每种情况提出了：（i）生成所需源多项式插值子的良态方法；（ii）多项式特解的高效转换公式。与对所有奇异和近奇异体积目标直接计算校正量相比，该方法将所有奇异和近奇异校正量推至边界区域附近目标点的近奇异层势评估，从而显著减少所需专用求积的数量。提供了正则化和求积近似的误差估计。该方法与成熟的快速算法兼容，不仅在在线阶段高效，而且设置过程也简便。数值算例证明了所提方法的高阶精度和效率。