题目：On continuous and discontinuous Galerkin approximations to second-kind Volterra integral equations

报告人：梁慧教授（哈尔滨工业大学深圳校区）

时间：2022年6月29日15:00-17:00

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摘要：Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces, in short, denoted as DC, CC, DG and CG methods respectively, are employed to solve second-kind Volterra integral equations (VIEs). It is proved that the quadrature DG and CG (QDG and QCG) methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas, are equivalent to the DC and CC methods, respectively. In addition, the fully discretised DG and CG (FDG and FCG) methods are equivalent to the corresponding fully discretised DC and CC (FDC and FCC) methods. The convergence theories are established for DG and CG methods, and their semi-discretised (QDG and QCG) and fully discretised (FDG and FCG) versions. In particular, it is proved that the CG method for second-kind VIEs possess a similar convergence to the DG method for first-kind VIEs. The above analysis is also extended to second-kind Volterra integral equations with weakly singualr kernels.

个人简介：

梁慧，哈尔滨工业大学（深圳）教授、博导。2008年7月获哈尔滨工业大学数学博士学位。2010.3.1-2011.9.31 在香港浸会大学担任客座研究学人，并多次访问香港浸会大学。2017.12.1-2018.11.30在加拿大纽芬兰纪念大学(Memorial University of Newfoundland) 担任访问学者。任SCI期刊Computational & Applied Mathematics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秘书。主要的研究方向为：延迟微分方程、Volterra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、深圳市基础研究计划、黑龙江省普通本科高等学校青年创新人才培养计划等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。目前共被SCI收录文章30余篇，发表在SIAM Journal on Numerical Analysis 、IMA Journal of Numerical Analysis、Journal of Scientific Computing、BIT Numerical Mathematics、Advances in Computational Mathematics、Applied Numerical Mathematics 等18种不同的国际杂志上。