Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems

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• 作者：Xiaoping Zhang^a ; ^{xpzhang.math@whu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Max Gunzburger^b ; ^{mgunzburger@fsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Lili Ju^c ; ^d ; ^{ju@math.sc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Collocation method ; Hypersingular integrals ; Nonlocal diffusion ; Nodal-type quadrature rules
• 刊名：Computer Methods in Applied Mechanics and Engineering
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：299
• 期：Complete
• 页码：401-420
• 全文大小：468 K

文摘

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.