报告人：沈捷 （厦门大学、美国Purdue大学）
报告题目：Efficient and accurate spectral methods for PDEs with singular solutions

报告摘要：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.

报告人简介：沈捷，美国普渡大学数学系教授、国际著名数值计算和分析专家。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。1991年起在宾西法尼亚州立大学任教，2002年转至普渡大学。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。因在数值分析、计算科学、计算流力力学和计算材料学等领域作出重要贡献, 沈捷教授2017年当选美国数学会Fellow. 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200余篇。

报告时间：2016年12月15日（星期四）15:00-16:00
报告地点：科技楼南楼702室