Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems
Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems
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DOI：
10.1016/j.cma.2015.11.008
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发表时间：
2016-02
影响因子：
7.2
通讯作者：
Xiaoping Zhang;M. Gunzburger;L. Ju
中科院分区：
文献类型：
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作者：
Xiaoping Zhang;M. Gunzburger;L. Ju
本文研究了逼近超奇异积分的积分型求积规则及其在有限部分积分方程和非局部扩散问题数值解中的应用。我们首先推导出明确的正交系数的表达式，并建立相应的误差估计。基于这些规则构造了一些配置格式，数值求解了某些类型的一维有限部积分方程和一维非局部扩散问题，并严格地得到了所提格式的条件数和最优误差估计.在均匀网格上，这些格式具有Toeplitz结构，这在开发快速线性求解器方面具有许多优点。各种数值实验也进行了说明的理论结果。
In this paper, we study nodal-type quadrature rules for approximating hypersingular integrals and their applications to numerical solution of finite-part integral equations and nonlocal diffusion problems. We first derive explicit expressions for the quadrature coefficients and establish corresponding error estimates. Some collocation schemes are then constructed based on these rules to numerically solve certain type of finite-part integral equations and nonlocal diffusion problems in one dimension, and condition number and optimal error estimates for the proposed schemes are also rigorously obtained. On uniform grids, these schemes are of Toeplitz structure which results in many advantages in developing fast linear solvers. Various numerical experiments are also performed to illustrate the theoretical results.