Multilevel augmentation methods for nonlinear boundary integral equations II: Accelerated quadratures and Newton iterations
Multilevel augmentation methods for nonlinear boundary integral equations II: Accelerated quadratures and Newton iterations
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DOI：
10.1216/jie-2012-24-4-545
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发表时间：
2012-12
影响因子：
0.8
通讯作者：
Xiangling Chen;Zhongying Chen;Bin Wu;Yuesheng Xu
中科院分区：
文献类型：
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作者：
Xiangling Chen;Zhongying Chen;Bin Wu;Yuesheng Xu
最近，同一作者提出了一种求解一类非线性边界积分方程的快速多层增广方法。在本文中，我们开发了加速求积公式计算中所涉及的MAM和近似迭代求解所产生的非线性系统的积分。具体来说，我们采用了产品的积分计划计算奇异积分出现在矩阵中所涉及的MAM和引入近似技术，在牛顿迭代求解所产生的非线性系统，以避免重复计算，在生成其雅可比矩阵。使用这两种技术的结果在一个修改的MAM，加快其计算。我们表明，修改后的MAM保留了原来的最佳收敛顺序，同时降低了计算成本。数值结果表明，与原算法和Atkinson和钱德勒的算法相比，改进的MAM算法具有更高的逼近精度和计算效率.
A fast multilevel augmentation method (MAM) was proposed recently by the same authors for solving a class of nonlinear boundary integral equations. In this paper, we develop accelerated quadrature formulas for computing the integrals involved in the MAM and approximate iteration for solving the resulting nonlinear system. Specifically, we employ a product integration scheme for computing the singular integrals which appear in the matrices involved in the MAM and introduce an approximation technique in the Newton iteration for solving the resulting nonlinear systems to avoid repeated computation in generating their Jacobian matrices. The use of these two techniques results in a modified MAM which speeds up its computation. We show that the modified MAM preserves the optimal convergence order of the original one while reducing computational costs. Numerical results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed modified MAM, with a comparison to those of the original one and a known algorithm of Atkinson and Chandler.