报告题目(Title)：LOWER EIGENVALUE BOUNDS FOR THE HARMONIC AND BI-HARMONIC OPERATOR (调和算子和双调和算子的特征值下界)
报告人 (Speaker)：Carsten Carstensen教授（德国洪堡大学）
报告时间 (Time)：2024年9月29日(周日) 14:00-15:00
报告地点 (Place)：校本部F309
邀请人(Inviter)：刘东杰 教授
主办部门：永利数学系
报告摘要：
Recent advances in the nonconforming FEM approximation of elliptic PDE eigenvalue problems include the guaranteed lower eigenvalue bounds (GLB) and its adaptive finite element computation. Like guaranteed upper eigenvalue bounds with conforming finite element methods,GLB arise naturally from the min-max principle, also named after Courant, Fischer, Weyl.The first part introduces the derivation of GLB for the simplest second-order and fourth-order eigenvalue problems with relevant applications,e.g., for the localization of in the critical load in the buckling analysis of the Kirchhoff plates. The second part studies an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace and bi-Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. The third part presents a modified hybrid high-order (HHO) eigensolver in the spirit of Carstensen,Ern, and Puttkammer [Numer. Math. 149, 2021] that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in a simplicial mesh.The error analysis allows for a priori quasi-best approximation and L2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs well in computer benchmarks with striking numerical evidence for optimal higher convergence rates.
