Finite Element Approximations of Singularities

讲座时间 Datetime: 

星期四, 2019/06/06 - 从 11:30 到 13:00

地点 Venue: 

海滨红楼6号楼1楼会议室

报告人 Speaker: 

李恒光 教授

单位 Affiliation: 

美国韦恩州立大学数学系

报告摘要 Abstract: 

Singular solutions occur in various mathematical models and can lead to significant challenges in theoretical justification and numerical approximations. In this talk, we first review the finite element formulation in a general mathematical setting. Then, we discuss recent advances in the development of effective finite element algorithms approximating a class of singular solutions, including corner singularities with different boundary conditions and singularities from the non-smooth points on the interface in transmission problems. We establish a-priori estimates (well-posedness, regularity, and the Fredholm property) for the singular solution in weighted Sobolev spaces. Then, based on these theoretical results, we propose a simple and explicit construction of the finite element space to recover the optimal convergence rate of the numerical solution. We also mention applications in physics and engineering.