一维矩形极坐标求积法及其误差分析
Rectangular polar quadrature in 1D and its error analysis
AI总结：
本文提出一维矩形-极坐标积分策略，结合Fejér第一求积法与多项式变量替换处理弱奇异卷积积分，推导了收敛条件与误差估计，数值实验验证了其在奇异积分方程及二维表面散射问题中的有效性。
摘要：
本文提出了矩形-极坐标（Rectangular-Polar, RP）积分策略的一维类似形式，并对弱奇异卷积积分进行了收敛性分析。该方法的核心思想是将整个积分拆分为多个不重叠子块（子域）上的积分，通过增加子块数量实现收敛，同时用固定数量的求积点精确逼近每个子块上的积分。对于非奇异积分，采用Fejér第一求积法实现高阶逼近；对于奇异积分，设计并引入了专门的积分策略，通过多项式变量替换（Polynomial Change of Variable, PCV）处理核的奇异性。我们证明，要实现高阶收敛，精确计算积分权重至关重要，且方法的收敛速率关键取决于PCV的次数和积分核的奇异性。具体而言，对于形式为$|x-y|^{-α}(α>-1)$的核，当且仅当$p(1-α) \in \mathbb{N}$时，该方法能实现高阶收敛，这一关系凸显了为PCV选择合适次数$p$的重要性。本文在“通过缩小逼近域尺寸、保持离散点数量固定实现收敛”的框架下，推导了Fejér第一求积法的新误差估计、切比雪夫系数的衰减速率，以及用离散切比雪夫系数逼近连续切比雪夫系数的误差。数值实验验证了理论结果，展示了RP策略在精确求解奇异积分方程方面的有效性。作为该方法的应用，本文计算了二维复杂域上表面散射问题的数值解，并验证了算法在大规模问题中的效能。
英文摘要：
This paper presents a one-dimensional analog of the Rectangular-Polar (RP) integration strategy and its convergence analysis for weakly singular convolution integrals. The key idea of this method is to break the whole integral into integral over non-overlapping patches (subdomains) and achieve convergence by increasing the number of patches while approximating the integral on patches accurately using a fixed number of quadrature points. The non-singular integrals are approximated to high-order using Fejér first quadrature, and a specialized integration strategy is designed and incorporated for singular integrals where the kernel singularity is resolved by mean of Polynomial Change of Variable (PCV). We prove that for high-order convergence, it is essential to compute integral weights accurately, and the method's convergence rate depends critically on the degree of the PCV and the singularity of the integral kernel. Specifically, for kernels of the form $|x-y|^{-α}(α>-1)$, the method achieves high-order convergence if and only if $p(1-α) \in \mathbb{N}$. This relationship highlights the importance of choosing an appropriate degree $p$ for the PCV. A new error estimate in a framework where convergence is achieved by reducing the approximating domain size while keeping the number of discretization points fixed is derived for the Fejér first quadrature, decay rate of Chebyshev coefficients, and the error in approximating continuous Chebyshev coefficients with discrete ones. Numerical experiments corroborate the theoretical findings, showcasing the effectiveness of the RP strategy for accurately solving singular integral equations. As an application of the method, numerical solutions of the surface scattering problem in the two dimensions are computed for complex domains, and the algorithm's efficacy is demonstrated for large-scale problems.
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