New 3D anisotropic mesh algorithms were proposed for the finite element method to approximate elliptic equations with singularities. These algorithms are simple, intuitive, and impose fewer geometric constraints on the domain. The resulting mesh is generally anisotropic and the associated numerical approximation has optimal convergence. In this talk, we report new developments in these algorithms both in 2D and 3D, especially sharp estimates on the condition numbers of the finite element stiffness matrices from these meshes.

Hengguang Li is now a Professor and Chair of Department of Mathematics of Wayne State University. His research is in the area of Applied Mathematics with main focus on Numerical Analysis, Partial Differential Equations, and Scientific Computing. Prof. Li’s work reflects an interplay of rigorous mathematical analysis of PDEs, the estimate and development of numerical methods, and their applications to physics, engineering, and medicine.