谱方法最新进展学术研讨会系列报告(2)

发布者:费洋发布时间:2019-11-22浏览次数:10

报告题目一:On the convergence of a two-level preconditioned Jacobi-Davidson method for eigenvalue problems

报告时间:2019112214:00-14:40

报告地点:理学院二楼会议室

报告人:许学军

报告人简介: 国家杰出青年基金获得者,同济大学数学科学学院院长,曾任中国计算数学学会常务理事、秘书长,国家973项目小组负责人,中国数学会理事,中国工业和应用数学会理事。曾获得过中科院优秀博士后称号和中国数学会钟家庆数学奖,以及中科院数学与系统科学研究院十大重大科研进展奖(2007)和中科院数学与系统科学研究院突出科研成果奖(2008)。在SIAM J. Numer. Anal.Numer. Math.等计算数学重要学术期刊上发表论文100余篇。

报告摘要:In this talk, we shall give a rigorous theoretical analysis of the two-level preconditioned Jacobi-Davidson method for solving the large scale discrete elliptic eigenvalue problems, which was essentially proposed by Zhao, Hwang, and Cai in 2016. Focusing on eliminating the error components in the orthogonal complement space of the target eigenspace, we find that the method could be extended to the case of the 2m th order elliptic operator (m = 1, 2). By choosing a suitable coarse space, we prove that the method holds a good scalability and we obtain the error reduction γ in each iteration, where C is a constant independent of the mesh size h and the diameter of subdomains Hδ = c[1-C(δ2m-1)/(H2m-1)] is the overlapping size among the subdomains, and c → 1 decreasingly as H 0. Moreover, the method does not need any assumption between H and h. Numerical results supporting our theory are given.

  

报告题目二:Interpolation and Expansion on Orthogonal Polynomials

报告时间:2019112214:40-15:20

报告地点:理学院二楼会议室

报告人:向淑晃

报告人简介:湖南省计算数学与应用软件学会理事长,中南大学教授,博士生导师。主要从事高振荡问题数值计算、矩阵理论及计算、线性互补问题、新古典主义数值分析等领域的研究。主持国家自然科学基金面上项目多项。在SIAM J. Numer. Anal.SIAM Sci. Comput.Math. Program 等国内外重要学术期刊发表论文100余篇。

报告摘要:The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this talk, new formulas on the convergence rates are considered. Moreover, new and optimal asymptotics on the coefficients of functions of limited regularity expanded in forms of Jacobi and Gegenbauer polynomial series are presented. All of these asymptotic analysis are optimal. Numerical examples illustrate the perfect coincidence with the estimates.

 

报告题目三:Spectral methods for some problems having low regularity solutions

报告时间:2019112215:20-16:00

报告地点:理学院二楼会议室

报告人:许传炬

报告人简介:福建省闽江学者特聘教授,厦门大学博士生导师。主要研究方向:偏微分方程数值分析,计算流体力学,反常扩散问题理论及计算。主持多项国家自然科学基金, 参加973项目、国家自然科学基金重点项目等。在SIAM J. Numer. Anal., SIAM J. Sci. Comput., Math. Comput.等计算数学重要学术期刊上发表论文100余篇。2003年获福建省科技进步二等奖,是国际计算数学顶级期刊SIAM J. Sci. Comput.的编委。

报告摘要:In this talk we will present a new spectral method for a class of equations with non-smooth solutions. The proposed method makes use of the fractional polynomials, also known as Muntz polynomials. We first present some basic approximation properties of the Muntz polynomials, including error estimates for the weighted projection and interpolation operators. Then we will show how to construct efficient spectral methods by using the Muntz polynomials. A detailed convergence analysis will be provided. The potential application of this method covers a large number of problems, including classical elliptic equations, integro-differential equations with weakly singular kernels, fractional differential equations, and so on.

 

报告题目四:Virtual element methods for elliptic variational inequalities of the second kind

报告时间:2019112216:00-16:40

报告地点:理学院二楼会议室

报告人:黄建国

报告人简介:上海交通大学教授,博士生导师,中国计算数学会理事。长期从事偏微分方程数值解,组合弹性结构问题的数学模型和有限元方法,反问题数值解等的研究工作。在SIAM J. Numer. Anal., SIAM J. Sci. Comput., Math. Comput. 等计算数学重要学术期刊上发表论文100余篇。先后主持多项国家自然科学基金项目,参加973项目等。

报告摘要:In this talk, we are concerned with virtual element methods for solving elliptic variational inequalities (EVIs) of the second kind. First, a general framework is provided for the numerical solution of the EVIs and for its error analysis. Then virtual element methods are applied to solve two representative EVIs: a simplified friction problem and a frictional contact problem. Optimal order error estimates are derived for the virtual element solutions of the two representative EVIs, including the effects of numerical integration for the non-smooth term in the EVIs. A fast solver is introduced to solve the discrete problems. Several numerical examples are included to show the numerical performance of the proposed methods. This is a joint with Fang Feng from Shanghai Jiao Tong University and Weimin Han from University of Iowa.


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