报告题目：MixedFinite Element Methods of Elasticity Problems
报告摘要：The problemsthat are most frequently solved in scientific and engineering computing mayprobably be the elasticity equations. The finite element method (FEM) wasinvented in analyzing the stress of the elastic structures in the 1950s. Themixed FEM within the Hellinger-Reissner (H-R) principle for elasticity yields adirect stress approximation since it takes both the stress and displacement asan independent variable. The mixed FEM can be free of locking for nearlyincompressible materials, and be applied to plastic materials, and approximateboth the equilibrium and traction boundary conditions more accurate. However,the symmetry of the stress plus the stability conditions make the design of themixed FEM for elasticity surprisingly hard. In fact, ``Four decades ofsearching for mixed finite elements for elasticity beginning in the 1960s didnot yield any stable elements with polynomial shape functions" [D. N.Arnold, Proceedings of the ICM, Vol. I : Plenary Lectures and Ceremonies(2002)]. Since the 1960s, many mathematicians have worked on this problem butcompromised to weakly symmetric elements, or composite elements. In 2002, usingthe elasticity complexes, Arnold and Winther designed the first family ofsymmetric mixed elements with polynomial shape functions on triangular grids in2D.
The talkpresents a new framework to design and analyze the mixed FEM of elasticityproblems, which yields optimal symmetric mixed FEMs. In addition, thoseelements are very easy to implement since their basis functions, based on thoseof the scalar Lagrange elements, can been explicitly written down by hand. Themain ingredients of this framework are a structure of the discrete stress spaceon both simplicial and product grids, two basic algebraic results, and atwo-step stability analysis method.
报告人简介：Jun Hu is a Professorof Mathematics at Peking University. His mainresearch interest is in finite element methods of partial differentialequations, including mixed finite element methods of problems arisingfrom mechanics, finite element methods of partial differential equationseigenvalue problems and high order prolems, adaptive finite element methods,and nonlinear approximations. In particular, he (with his collaorators)solved a difficult problem: constructions of stable mixedfinite elements of elasticity problems within the Hellinger-Reissnerformulation. He has published over 60 publications in peer-reviewed journals,and serves as President of the Beijing Society of ComputationalMathematics, one of managing editors of journal Advances in AppliedMathematics and Mechanics, the editor of three journals: ComputationalMethods in Applied Mathematics, Computer and Mathematics withApplications, and Journal of Computational Mathematics. He is therecipient of National Science Fund for Distinguished Young Scholarof PR China, 2016, the first Youth Innovation prize of China Society forComputational Mathematics in 2015 and the National Excellent DoctoralDissertation of PR China in 2006, and a former Alexander von Humboldt ResearchFellow of Alexander von Humboldt Foundation of Germany in 2004.