活動主題：計算數(shù)學及其交叉學科前沿系列講座報告
報告題目：Finite element methods for the perfect conductivity problem with close-to-touching inclusions
報告人：楊宗澤 博士后 香港理工大學
邀請人：董灝博士
報告時間：2024年5月24日上午9:00-12:00
騰訊會議ID：440-644-993
報告人簡介：楊宗澤, 博士. 香港理工大學博士后, 合作導師為香港理工大學李步揚教授. 2020年于西北工業(yè)大學獲得博士學位. 2018年12月至2019年12月于澳大利亞昆士蘭科技大學進行聯(lián)合培養(yǎng). 2020年7月至2021年9月于深圳京魯計算科學應(yīng)用研究院進行博士后研究工作. 研究方向為曲面PDE的能量遞減算法, 移動界面問題的ALE有限元方法, 分數(shù)階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等計算數(shù)學知名期刊發(fā)表論文十余篇.
報告摘要：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.
主辦單位：數(shù)學與統(tǒng)計學院
