报告题目：Locally Conforming Immersed Finite Element Spaces for Solving Interface Problems via DG

报告人：林涛 教授

报告时间：2026年5月20日（星期三）上午10:20-11:50

报告地点：文科综合楼E-2029

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科研处  学科办  数理学院（应用数学研究所）

2026年05月15日

摘要: 

In solving interface problems of partial differential equations, Immersed Finite Element (IFE) methods stand out for their ability to work on interface-independent meshes, by utilization of standard finite element functions on non-interface elements, together with IFE functions applied specifically on interface elements. However, IFE functions proposed in the literature, even though designed to adhere to interface jump conditions, are not conforming to the underlying Sobolev spaces. To mitigate this discrepancy, penalties are commonly employed in the related IFE methods, albeit at the cost of making these IFE methods less desirable. In this presentation, we introduce a new class of IFE functions, aiming to overcome the aforementioned shortcoming. Our approach is to employ the Frenet-Serret apparatus from the differential geometry of the interface curve to establish an orthogonal curvilinear coordinate system in the vicinity of the interface. Each shape function in the local IFE space constructed in this method is a composition of a piecewise Q^m polynomial taking care of the interface jump conditions in Frenet coordinates, and the Frenet transformation representing essential features of the differential geometry of the interface. Consequently, such IFE functions are locally conforming to the underlying Sobolev space of the interface problem, especially suitable for deployment in standard DG finite element schemes. We will provide numerical results to illustrate features of IFE functions generated through this novel methodology.

报告人简介: 

林涛，弗吉尼亚理工大学教授、博士生导师。1990年在怀俄明大学取得博士学位，1989年在弗吉尼亚理工大学数学系受聘为助理教授，2001年起担任教授。他是计算数学和科学计算方面的专家，研究兴趣涉及偏微分方程和积分微分方程的数值求解方法，尤其是在界面问题的浸入式有限元方法及其应用方面取得了开创性的研究成果。在SIAM Journal on Numerical Analysis、Numerische Mathematik和Journal of Computational Physics等国际顶级期刊发表研究性论文100余篇，主持多项美国自然科学基金的科研项目。