论文标题
关于通过密度插值法和应用程序正规化Cauchy型积分运算符
On the regularization of Cauchy-type integral operators via the density interpolation method and applications
论文作者
论文摘要
本文介绍了一种正规化技术，用于对几乎单数，主要价值和有限的零件cauchy型积分运算符的高阶有效数值评估。通过依靠凯奇公式，凯奇 - 古尔萨特定理以及输入密度的曲线泰勒插值，提出的方法可以将凯奇和相关的积分运算符重塑为平滑的轮廓积分。因此，可以通过基本的正交规则在复杂平面的任何地方进行准确评估，包括附近和轮廓的有问题点。该技术在评估拉普拉斯层电位和相关积分运算符以及计算共形映射的应用中的应用。尤其是前一种应用，比最近引入的谐波密度插值方法的显着改善。介绍了光滑和分段光滑轮廓的光谱准确的离散方法。各种数值示例，包括解决弱奇异和超级拉普拉斯边界积分方程的解决方案，以及对具有挑战性的保形映射的评估，证明了在这种情况下密度插值方法的有效性和准确性。
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.
