The superconvergence of the composite midpoint rule for the finite-part integral
The superconvergence of the composite midpoint rule for the finite-part integral
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DOI：
10.1016/j.cam.2009.09.030
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发表时间：
2010-02
期刊：
影响因子：
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通讯作者：
Jiming Wu;Zihuan Dai;Xiaoping Zhang
中科院分区：
文献类型：
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作者：
Jiming Wu;Zihuan Dai;Xiaoping Zhang
复合中点法则可能是黎曼积分的Newton-Cotes法则中最简单的一个。然而，这一规则是发散的一般阿达玛有限部分积分。本文将这一规则转化为一个有用的规则，并应用它来计算Hadamard有限部分积分以及求解相应的积分方程。重点是研究其逐点超收敛现象，即，当奇异点与某个先验已知点重合时，中点法则的收敛速度比全局可能收敛速度要快。证明了复合中点规则的超收敛速度发生在每个子区间的中点处，并得到了相应的超收敛误差估计。利用中点法则逼近有限部分积分，并选择超收敛点作为配置点，得到了求解有限部分积分方程的配置格式.更有趣的是，所得到的线性系统的系数矩阵的逆有一个显式的表达式，由它建立的最佳误差估计。数值算例验证了理论分析的正确性。
The composite midpoint rule is probably the simplest one among the Newton–Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis.