Speaker: Dr. Xu Zhang (张旭博士)

Time: 12:10-13:10, 22 December 2023 (Friday) (Beijing time)

Venue: A103,Lijiao Building

Tencent Meeting ID: 433-535-918

Abstract

Interface problems arise in many applications in science and engineering. Partial differential equations (PDEs) are often used to model interface problems. Solutions to these PDE interface problems often involve kinks, singularities, discontinuities, and other non-smooth behaviors, which often pose challenges in obtaining satisfactory solutions. The immersed finite element method (IFEM) is a class of numerical methods for solving PDE interface problems with unfitted meshes.

In this talk, I will present recent advances in the development and analysis of IFEM for solving 3D interface problems with general non-homogeneous jump conditions. The proposed method can be utilized on interface-unfitted tetrahedral and cuboidal meshes, even if the 3D interface surface has an arbitrary shape. Distinguished from many unfitted-mesh methods in the literature, our method follows the homogenization idea, and the enrichment functions are immediately determined by the jump data, which does not introduce any additional degrees of freedom to the system, but still leads to optimal convergence. The resulting homogenized IFE space is isomorphic to the standard finite element space on the same mesh, and the isomorphism is stable with respect to the interface location. This feature leads to the optimal h^2 conditioning independent of interface location, which allows us to build an efficient multigrid grid solver. Numerical examples will be provided not only to validate our theoretical results, but also to demonstrate the applicability of the method when dealing with some real-world 3D interface models.

About Dr. Zhang

Xu Zhang is an Associate Professor in the Department of Mathematics at Oklahoma State University. He received his B.S. degree (2005) and M.S. degree (2008) in Computational Mathematics from Sichuan University, and his Ph.D. degree (2013) in Mathematics from Virginia Tech. He did postdoc training at Purdue University during 2013-2016. Before joining OSU, he was an Assistant Professor at Mississippi State University during 2016-2019. His research is on numerical methods for partial differential equations. Recently, his research focused on immersed finite element methods for PDE interface problems including algorithm development, implementation, error analysis, and engineering applications. He has authored over 30 journal articles published in SINUM, SISC, JCP, CMAME, JSC, etc., and has been cited more than 1100 times, according to Google Scholar. He has received multiple research grants from National Science Foundation.