报告人：张智民（韦恩州立大学）

报告题目：How well can we hear the shape of a drum by computer algorithms?

报告摘要：Can we determine the shape of a domain by its Laplacian eigenvalues? The question puzzled us for many years until 1992, when three mathematicians surprised everyone with a counterexample. However, this is not the end of the story to applied mathematicians, since in most cases we are unable to obtain exact eigenvalues, and numerical approximation by computer algorithms is necessary. Naturally, another question arises: How many numerical eigenvalues can we trust? When approximating PDE eigenvalue problems using numerical methods such as finite difference and finite element, it is commonly known that only a small portion of numerical eigenvalues are reliable. However, this knowledge is tipycally qualitative rather than quantitative in the literature.In this talk, I will first survey some theoretical results from pure mathematics regarding eigenvalue problems. Then I will investigate the number of “trusted” eigenvalues by the finite element (and the related finite difference method results obtained from mass lumping) for approximating 2mth order elliptic PDE eigenvalue problems. Our two model problems are the Laplace and bi-harmonic operators, for which solid knowledge regarding the magnitudes of eigenvalues is available in the literature. By combining this knowledge with a priori error estimates of the finite element method, we can roughly determine how many “reliable” eigenvalues can be obtained from numerical approximation under a predetermined convergence rate.

报告时间：2024年10月25日（星期五）19:00-22:00

报告地点：科技楼南706室

邀请人：李东方

报告人简介：张智民，美国韦恩州立大学教授，Charles H. Gershenson 杰出学者。研究方向是偏微分方程数值解，包括有限元，有限体积，谱方法等，发表学术论文200余篇；提出的多项式保持重构Polynomial Preserving Recovery（PPR）格式于2008年被国际上广为流行的大型商业软件 COMSOL Multiphysics 采用，并使用至今。担任或曾任“Mathematics of Computation” “Journal of Scientific Computing” 等9个国际计算数学杂志编委。