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發布時間:2020-12-02【告訴好友】 【關閉窗口】

  報告時間:2020.12.10 10:00-12:30

  會議主題:貽民預定的會議  會議時間:2020/12/10 09:30-13:00


  會議 ID:541 336 924   會議密碼:123456


  報告題目:Babuska Problem in Composite Materials and its Applications

  報告摘要:A long-standing area of materials science research has been the study of electrostatic, magnetic, and elastic fields in composite with densely packed inclusions whose material properties differ from that of the background. For a general elliptic system, when the coefficients are piecewise Holder continuous and uniformly bounded, an ε-independent bound of the gradient was obtained by Li and Nirenberg, where ε represents the distance between the interfacial surfaces. However, in high-contrast composites, when ε tends to zero, the stress always concentrates in the narrow regions. As a contrast to the uniform boundedness result of Li and Nirenberg, in order to investigate the role of ε played in such kind of concentration phenomenon, in this talk we will show the blow-up asymptotic expressions of the gradients of solutions to the Lame system with partially infinite coefficients in dimensions two and three. This completely solves the Babuska problem on blow-up analysis of stress concentration in high-contrast composite media. Moreover, as a byproduct, we establish an extended Flaherty-Keller formula on the effective elastic property of a periodic composite with densely packed fibers, which is related to the “Vigdergauz microstructure” in the shape optimizition of fibers.


  李海剛,教授,博士生導師。2007年國家建設高水平大學首批公派研究生,北京師范大學(導師:保繼光教授)與美國羅格斯(Rutgers)大學(導師:李巖巖教授)聯合培養博士。主要研究來自材料力學和幾何學中的線性和非線性偏微分方程理論。在復合材料中Lame方程組解的梯度估計(Babuska問題)和Monge-Ampere方程、Hessian方程的外Dirichlet問題等方面做出一系列深刻的原創性成果,在《Adv.Math.》(2篇)、《Arch. Ration. Mech. Anal.》(2篇)、《Trans. Amer. Math. Soc.》、《Calc. Var. Partial Differential Equations》、《SIAM J.Math. Anal.》、《J. Differential Equations》等SCI國際主流數學雜志上發表科研論文20余篇。2013年獲得“京師英才”一等獎。2014年8月在韓國舉行的國際數學家大會(ICM2014)衛星會議上做邀請報告。2015年8月在北京舉行的國際工業與應用數學大會(ICIAM2015)Minisymposia做邀請報告。2016年獲得教育部霍英東教育基金會第十五屆高等院校青年教師基金。(兩年一屆,每屆數學學科僅資助4人)。2018年2月獲得教育部自然科學二等獎。


  報告題目:Super-resolution property of splitting methods for Dirac equation in the nonrelativistic limit regime

  摘要:We establish error bounds of the Lie-Trotter splitting and Strang splitting for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials. In this regime, the solution admits high frequency waves in time. Surprisingly, we find out that the splitting methods exhibit super-resolutions,  i.e. the methods can capture the solutions accurately even if the time step size is much larger than the sampled wavelength. Lie splitting shows half order uniform convergence w.r.t temporal wave length. Moreover, if  the time step size is non-resonant, Lie splitting would yield an improved uniform  first order uniform error bound. In addition, we show Strang splitting is uniformly convergent with half order rate for general time step size  and uniformly convergent with three half order rate for non-resonant time step size. Finally, numerical examples are reported to validate our findings.

  蔡勇勇,教授,博士生導師。本科和碩士就讀于北京大學數學科學學院,2012年于新加坡國立大學數學系獲博士學位,后至威斯康辛大學麥迪遜分校、馬里蘭大學帕克分校和普渡大學從事博士后研究工作,2016年至2019年期間任北京計算科學研究中心特聘研究員。主要研究偏微分方程的數值方法及其在量子力學等領域中的應用。相關研究結果發表在SIAM Journal on Numerical Analysis、SIAM Journal on Applied Mathematics、SIAM Journal on Mathematical Analysis、Mathematics of Computation、Journal of Computational Physics、Journal of Scientific Computing、Journal of Functional Analysis等學術期刊上。