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报告题目：Finite element methods for the perfect conductivity problem with close-to-touching inclusions

报告人：杨宗泽 博士后 香港理工大学

邀请人：董灏博士

报告时间：2024年5月24日上午9:00-12:00

腾讯会议ID：440-644-993  

报告人简介：杨宗泽, 博士. 香港理工大学博士后, 合作导师为香港理工大学李步扬教授. 2020年于西北工业大学获得博士学位. 2018年12月至2019年12月于澳大利亚昆士兰科技大学进行联合培养. 2020年7月至2021年9月于深圳京鲁计算科学应用研究院进行博士后研究工作. 研究方向为曲面PDE的能量递减算法, 移动界面问题的ALE有限元方法, 分数阶微分方程的有限元方法. 目前已在SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computation, Journal of Computational Physics等计算数学知名期刊发表论文十余篇.

报告摘要：In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $\varepsilon$ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $\varepsilon$ between the inclusions, in both two and three dimensions for general convex smooth inclusions which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.

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