报告人：李步扬教授（香港理工大学）

报告地点：#腾讯会议：449-452-983

报告时间：2026年8月31日16：30

题目：Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions

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 still 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. In particular, we propose a special finite element basis function which resolves the asymptotic singularity of the solution by making the interpolation error bounded in W^{1,∞} in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in W^{1,∞}. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the H¹ norm, uniformly with respect to the distance ε 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.

报告人简介：李步扬, 香港理工大学应用数学系讲席教授，国家杰出青年科学基金项目获得者，主要研究领域为偏微分方程的数值计算和数值分析，包括曲率流的参数化有限元逼近、非线性色散和波动方程不光滑解的计算、不可压Navier–Stokes 方程的数值解及其误差分析、高频 Helmholtz 方程的有限元和 PML 方法、非线性抛物方程、相场方程、分数阶偏微分方程、Ginzburg-Landau 超导体方程、热敏电阻方程的数值分析，以及有限元法、谱方法、Convolution quadrature等，现为计算数学杂志 Mathematics of Computation、SIAM Journal on Numerical Analysis、Numerische Mathematik、IMA Journal of Numerical Analysis等杂志编委。