题 目：On convergence of the immersed boundary method for elliptic interface problem and the augmented IIM using pressure formulation with traction interface conditions
报告人：Professor Zhilin Li
(North Carolina State University, USA)
时 间：7月10日（周二）， 16:00--17：00
地 点：数学楼210室
报告摘要： Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin's IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{/infty}$ norm. But almost none rigorous proof could be found in the literature until recently by Mori who showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show the first order convergence with a $/log h$ factor of the IB method for elliptic interface problems essential without the boundary condition restrictions. The results should be applicable to the IB method for many different situations involving elliptic solvers for Stokes and Navier-Stokes equations.
The second part of my talk is about recent progress of the augmented IIM using pressure formulation with traction interface conditions. The crucial part is how to to approximate the Laplacian for pressure boundary condition prediction which is easy for rectangular domain but difficult for arbitrary interface. We show the accuracy and the stability of the numerical method.
专家简介：李治林，北卡莱罗纳州立大学数学系教授（江苏特聘教授）,曾主持、参与美国国家自然科学基金(NSF)项目、美国陆军研究办公室基金(ARO)项目、美国空军研究办公室基金(AFOSR)项目等，1991年曾获得院长奖学金，1997年获得Oak Ridge 联合大学青年优秀教师奖，他创造性地提出浸入界面法（IIM）、浸入有限元法(IFEM)以及快速浸入界面法(AIIM)等一系列有效求解涉及含有奇异源项、不连续的物理参数及不规则区域上、自由边界、移动界面等问题的偏微分方程的数值方法，并有效模拟了许多计算流体力学方面的问题。先后发表学术论文75余篇，2006年出版了题为“浸入界面法：数值求解含有界面以及在不规则区域上的偏微分方程”的专著。他先后组织多次国际学术会议，指导过8个博士研究生、多个硕士研究生和博士后。他的研究领域包括：数值分析和科学计算；含有自由界面和移动界面的偏微分方程数值方法研究，不规则区域问题求解，有限差分方法及有限元方法研究，计算流体力学，生物流体力学及其应用。
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