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【明理讲坛】数学中心“椭圆型偏微分方程与非线性泛函分析”报告会
发布时间:2020-12-02【告诉好友】 【关闭窗口】

  报告时间:2020.12.10 10:00-12:30

  会议主题:贻民预定的会议  会议时间:2020/12/10 09:30-13:00

  链接:https://meeting.tencent.com/s/dnHTyrfqa8A1

  会议 ID:541 336 924   会议密码:123456

  (一)报告人:李海刚北京师范大学

  报告题目:Babuska Problem in Composite Materials and its Applications

  报告摘要: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.

  报告人简介:

  李海刚,教授,博士生导师。2007年国家建设高水平大学首批公派研究生,北京师范大学(导师:保继光教授)与美国罗格斯(Rutgers)大学(导师:李岩岩教授)联合培养博士。主要研究来自材料力学和几何学中的线性和非线性偏微分方程理论。在复合材料中Lame方程组解的梯度估计(Babuska问题)和Monge-Ampere方程、Hessian方程的外Dirichlet问题等方面做出一系列深刻的原创性成果,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国际主流数学杂志上发表科研论文20余篇。2013年获得“京师英才”一等奖。2014年8月在韩国举行的国际数学家大会(ICM2014)卫星会议上做邀请报告。2015年8月在北京举行的国际工业与应用数学大会(ICIAM2015)Minisymposia做邀请报告。2016年获得教育部霍英东教育基金会第十五届高等院校青年教师基金。(两年一届,每届数学学科仅资助4人)。2018年2月获得教育部自然科学二等奖。

  (二)报告人:蔡勇勇北京师范大学)

  报告题目: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,博士生导师本科和硕士就读于北京大学数学科学学院,2012年于新加坡国立大学数学系获博士学位,后至威斯康辛大学麦迪逊分校、马里兰大学帕克分校和普渡大学从事博士后研究工作2016年至2019年期间任北京计算科学研究中心特聘研究员。主要研究偏微分方程的数值方法及其在量子力学等领域中的应用。相关研究结果发表在SIAM Journal on Numerical AnalysisSIAM Journal on Applied MathematicsSIAM Journal on Mathematical AnalysisMathematics of ComputationJournal of Computational PhysicsJournal of Scientific ComputingJournal of Functional Analysis等学术期刊上。

      
      
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