梁慧 教授（哈尔滨工业大学（深圳））

龚伟 研究员、毛士鹏 研究员

Compatibility of collocation error analyses for Volterra integral equations with smooth and weakly singular kernels

5月21日（周四）10:00-11:00，思源楼817

The numerical solution by piecewise polynomial collocation and iterated collocation of Volterra integral equations (VIEs) of the second kind has been extensively studied and apparently sharp convergence results are known for the cases of a smooth kernel $K(t,s)$ and a weakly singular kernel $(t-s)^{-\alpha}K(t,s)$, where $\alpha\in (0,1)$ is a parameter.If one takes the formal limit as $\alpha\to 0$, then the weakly singular VIE reduces to the smooth VIE, but the known collocation error bounds for the weakly singular VIE do not become the collocation error bounds for the smooth VIE --- the error bounds for the smooth VIE are typically of a higher order. In the current paper this anomaly is explained and new sharper collocation and iterated collocation error bounds are derived for the weakly singular VIE  that blend exactly as $\alpha\to 0$ with known error bounds for the smooth VIE. This analysis is substantially different from previous VIE collocation analyses, e.g., it constructs a remarkable new decomposition of the solution of the weakly singular VIE, it investigates in detail the dependence on $\alpha$ of the matrices associated with collocation, it establishes a new Gronwall inequality, and the dependence of the error on the parameter $\alpha$ is traced precisely throughout the work. Numerical experiments are presented to illustrate our theoretical results.

报告人简介：梁慧，哈尔滨工业大学（深圳）教授、博导。入选首届“深圳市优秀科技创新人才培养项目（杰出青年基础研究）”。2008年7月获哈尔滨工业大学数学博士学位。任期刊《Computational & Applied Mathematics》、《Journal of Integral Equations and Application》和《Communications on Analysis and Computation》编委，中国仿真学会仿真算法专委会委员、中国仿真学会不确定性系统分析与仿真专业委员会副主任委员兼秘书、广东省数学会常务理事等。主要研究方向为：微分及积分方程的数值分析。主持国家自然科学基金面上项目、青年项目、深圳市杰出青年基金项目、深圳市基础研究计划等10余项科研项目，获中国系统仿真学会“2015年优秀论文”奖、2018第二届黑龙江省数学会优秀青年学术奖、深圳市海外高层次人才认证、吉林省自然科学一等奖（排名第五）。在SIAM J.Numer. Anal.、IMA J. Numer. Anal.、J. Sci. Comput.、BIT等期刊发表学术论文50余篇。