This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on the Cauchy formula, the Cauchy-Goursat theorem, and on-curve Taylor interpolations of the input density, the proposed methodology allows to recast the Cauchy and associated integral operators as smooth contour integrals. As such, they can be accurately evaluated everywhere in the complex plane -- including at problematic points near and on the contour -- by means of elementary quadrature rules. Applications of the technique to the evaluation of the Laplace layer potentials and related integral operators, as well as to the computation conformal mappings, are examined in detail. The former application, in particular, amounts to a significant improvement over the recently introduced harmonic density interpolation method. Spectrally accurate discretization approaches for smooth and piecewise smooth contours are presented. A variety of numerical examples, including the solution of weakly singular and hypersingular Laplace boundary integral equations, and the evaluation of challenging conformal mappings, demonstrate the effectiveness and accuracy of the density interpolation method in this context.
翻译:本文介绍了对近单一、本值和有限成份的Cauchy型整体操作员进行高顺序高效数字评估的正规化技术。通过依赖Cauchy公式、Cauchy-Goursat理论以及输入密度的在曲线泰勒内推法,拟议方法允许将Cauchy和相关整体操作员重新定位为光滑的轮廓组合体。因此,在复杂的平面上任何地方 -- -- 包括在问题点附近和轮廓上 -- -- 都可以通过基本二次曲线规则来准确评估它们。对Laplace层潜力和相关整体操作员的评价以及计算符合性绘图技术的应用进行了详细研究。前一种应用,特别是相当于最近引入的调和密度内插法的显著改进。介绍了对光滑和小片光滑的轮廓的精确分解方法。各种数字实例,包括弱小单和超超超超拉普特边界组合方程式的解决方案,以及具有挑战性的符合性准度的剖面图的评估工作,展示了这一方法的有效性和准确性。