AI 中文总结
针对科学计算中奇异和近奇异积分评估难题，尤其是高维情况，本文基于高斯超几何函数变量变换技术，提出系统方法，经分析和实验验证，该方法能指数收敛，优于直接高斯 - 勒让德求积法，为高维近奇异积分提供实用框架。
AI 中文摘要
在科学计算中，对具有奇异或近奇异核的积分进行精确数值评估仍是一项基本挑战，尤其是随着空间维度增加。现有方法在一维和二维取得一定成功，但扩展到高维常受近奇异行为及数值不稳定性阻碍。本文基于高斯超几何函数导出的变量变换技术，提出一种在任意维度下高精度评估奇异和近奇异积分的系统方法。分析了达菲域和物理笛卡尔域中奇点和近奇点结构，通过双曲正弦变换进行统一处理。基于伯恩斯坦椭圆参数的收敛分析表明，双曲正弦变换（β = 1）优于其他超几何变换（β = 2, 3）。二维到五维的数值实验及变分离散中求积诱导一致性误差分析表明，该方法实现指数收敛且大幅优于直接高斯 - 勒让德求积法。该方法为高维近奇异积分提供了实用且理论基础扎实的框架，在科学计算中有广泛适用性。
英文摘要
The accurate numerical evaluation of integrals with singular or nearly singular kernels remains a fundamental challenge in scientific computing, particularly as the spatial dimension increases. While existing approaches, including singularity extraction, adaptive element subdivision, and analytical integration schemes, have achieved considerable success in one and two dimensions, their extension to higher dimensions is often hindered by severe near-singular behaviour and the resulting numerical instabilities. In this work, we propose a systematic approach for the high-accuracy evaluation of singular integrals and nearly singular integrals in arbitrary dimensions, based on variable transformation techniques derived from the Gauss hypergeometric function. We further analyze the structure of singularities and near-singularities in both the Duffy domain and the physical Cartesian domain, and establish a unified treatment via the sinh transformation. A convergence analysis based on the Bernstein ellipse parameter reveals that the sinh transformation ($β=1$) consistently outperforms alternative hypergeometric transformations ($β=2,3$), which introduce secondary near-singularities despite achieving formal cancellation. Numerical experiments in dimensions two through five, together with an analysis of quadrature-induced consistency errors in variational discretisations, demonstrate that the proposed approach achieves exponential convergence and substantially outperforms direct Gauss--Legendre quadrature. The method offers a practical and theoretically grounded framework for high-dimensional nearly singular integration with broad applicability in scientific computing.
