哈工大首届数学冬季学校和论坛

——谱方法数值分析研讨会

研讨会时间定在1月5－8日.

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报告1：

Title: Efficient and accurate spectral methods for PDEs with singular solutions

Author: Jie Shen, Xiamen University and Purdue University

Abstract: The usual spectral methods will provide high-order accuracy for problems with smooth solutions. However, they may not work well for problems with singular solutions due to various facts such as corner singularities, non-matching boundary conditions, non-smooth coefficients. If the form of the singular expansion for the solution is known, we develop a Muntz Galerkin method which is based on specially tuned Muntz polynomials to deal with the singular behaviors of the underlying problems, and show that it provide optimal error estimates.  On the other hand, if the Muntz Galerkin method is not applicable or efficient, we present a new extended spectral-Galerkin method which allows us to split it into two separate problems: one is to find an approximation for the smooth part by a usual spectral method, the other is to determine an approximation to the singular part with $k$ terms by solving a $k/times k$ system. So the new method is very easy to implement, very efficient and is capable of providing very accurate approximations for a class of singular problems.  We will present ample numerical results for a variety of problems with singular solutions, including fractional PDEs, to demonstrate the effectiveness of our approaches.

报告时间：2017年1月8日（周日）8:30—9:30

报告地点：格物楼503

报告人简介：沈捷教授，美国普渡大学数学系教授。1982年毕业于北京大学计算数学专业，随后赴法国巴黎十一大学研究数值分析，师从国际著名数学大师R.TEMAM,1987年获得博士学位后赴美在Indiana University从事博士后研究。沈捷教授是国际著名的数值数学家，主要从事偏微分方程数值解研究，特别在谱方法数值分析理论和科学计算方面有杰出贡献，同时在海洋和大气动力系统以及材料科学计算方面也有很深的造诣，经常在重要国际学术会议上作大会主报告。 2008年沈捷教授因在微分方程研究中的卓越贡献获得富布赖特奖。沈捷教授长期从事偏微分方程数值解的研究，尤其在谱方法和投影法上有很多杰出的工作，在SIAM.J.Numer.Anal., SIAM.J.Sci.Comput., Numer.Math.，Math.Comp.等国际著名期刊上发表学术论文100余篇。

报告2：

Title: high order methods for abnormal diffusion equations

Author: Chuanju Xu, Xiamen University

Abstract: In this talk, we discuss high order methods for time fractional diffusion equations. The methods use high order finite differences to approximate the fractional derivative in time, resulting in a time-stepping scheme for the underlying equation. Then the resulting equations are discretised in space by using spectral methods. The main body of this talk is devoted to carry out a rigorous analysis for the stability and convergence of the proposed methods.

报告时间：2017年1月8日（周日）9:30—10:30

报告地点：格物楼503

报告3：

Title: A Fractional Phase-Field Model and Related Equations

Author: Chuanju Xu (Xiamen University)

Abstract: In the first part of this talk we discuss a fractional mass-conserving Allen-Cahn phase-field model that describes the mixture of two incompressible fluids. The new fractional model allows controlling the sharpness of the interface, which is typically diffusive in integer-order phase-field models. The spatial discretization is based on a Galerkin spectral method whereas the temporal discretization is based on a stabilized ADI scheme. A number of numerical examples are provided to demonstrate the accuracy of the method and the ability to control the interface thickness between two fluids. In the second part, we will discuss about theoretical development of the fractional Navier-Stokes equations.

报告人简介：许传炬教授，厦门大学数学科学学院教授，博士生导师。1988年到法国攻读博士学位。1989年从法国巴黎南大学计算数学硕士毕业，1993年获巴黎第六大学计算数学博士学位，同年回国工作。先后多次前往法国，丹麦，美国，香港和新加坡访问。主要研究方向：计算流体力学中的高阶算法研究和湍流LES模拟。主持国家自然科学基金2项, 参加973项目1项, 参加国家自然科学基金重点项目1项,主持国家留学回国基金1项, 福建省自然科学基金3项, 科技部 "中法先进研究计划"课题1项。2000年获法国法中科技协会颁发的信息科学2000奖。2003年获福建省科技进步二等奖。入选教育部新世纪优秀人才培养计划。