This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green’s function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green’s function. For the sake of definiteness, we focus here on Nystr‘̀om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method’s accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples.

本文提出了一种通用的高阶核正则化技术，适用于在二维和三维空间维度上与线性椭圆PDE关联的Calderón演算的所有四个积分算子。像以前的密度插值方法一样，所提出的技术依赖于根据底层齐次PDE的解在内核奇异点周围插值密度函数，从而根据有界（或更规则）的被积数重铸奇异和几乎奇异的积分。我们在这里提出一种简单的插值策略，与以前的方法不同，该策略不需要对沿表面的密度函数的高阶导数进行显式计算。此外，在寻求密度的意义上，所提出的方法与内核和尺寸无关插值法构造成点源场的线性组合，由积分方程公式中使用的格林函数给出，从而使该程序原则上适用于任何已知格林函数的PDE。为了确定起见，我们在这里集中于Nystr'̀om方法（标量）的Laplace和Helmholtz方程以及（矢量）的弹性和时谐弹性动力学方程。通过各种大型三维数值示例，证明了该方法的准确性，灵活性，效率以及与快速求解器的兼容性。