荔园数学讲坛: High Accurate Algorithms for Eigenvalue Problems Based on Nonconforming Finite Element Methods

主讲人：北京大学 胡俊 教授

时间：2023年3月7日15:00–16:00

地点：腾讯会议：623-821-985 密码：1234

会议链接：https://meeting.tencent.com/dm/jwJtgnuxJQFN

内容摘要： 

This talk presents high accurate algorithms including the extrapolation methods and the postprocessing algorithms, based on the Crouzeix-Raviart element and the enriched Crouzeix-Raviart element for second order elliptic problems and the Morley element for fourth order elliptic problems, for eigenvalue problems. The superconvergence analysis of eigenvalues by the extrapolation methods is based on an optimal asymptotic expansion of discrete eigenvalues under consideration. The canonical interpolation of these triangular nonconforming finite elements lack the crucial superclose property, which leads to a significant difficulty in both the asymptotic analysis and analysis of the postprocessing algorithms. For these three nonconforming finite elements, based on some equivalence between the nonconforming finite elements and the corresponding mixed finite elements,  a required asymptotic result for the extrapolation methods is deduced from the superclose property of the associated mixed finite elements. Meanwhile, two asymptotically exact a posteriori error estimators are proposed, one is derived from the canonical interpolation of the nonconforming finite elements under consideration while another one is derived from that of the continuous linear finite element. Further, these a posteriori error estimators are employed to construct and analyze postprocessing algorithms for the eigenvalue problems.

主讲人简介: 

胡俊，北京大学数学科学学院教授。主要从事非标准有限元方法的研究，建立了一个设计线弹性力学问题混合有限元方法的新框架, 构造了以多项式为形函数、应力严格对称、有最优收敛性的稳定混合有限元，彻底解决了线弹性力学问题混合有限元的构造这个长期悬而未决的难题；首次构造出线性化Einstein-Bianchi方程组对称的稳定混合有限元。曾获国家杰出青年科学基金、冯康科学计算奖、中国数学会计算数学分会首届青年创新奖。 现任Adv Appl. Math. Mech.执行主编、北京计算数学学会理事长、和中国数学会常务理事。