In this project, Dr. Li proposes a systematic research on finite element methods (FEMs) for singular solutions of elliptic partial differential equations (PDEs), especially in 3D. Targeting fundamental theoretical and numerical issues, this research has two main components. (I) Innovative numerical advancements: the development of new 3D meshing algorithms. Simple, explicit, and well structured, these meshes can effectively capture the local behavior of the singular solution and lead to optimal FEMs. (II) Rigorous theoretical investigations: (i) sharp regularity estimates in new function spaces; (ii) sharp error analysis in energy and non-energy norms on both 2D graded meshes and the proposed 3D meshes; (iii) fast multigrid-based numerical solvers on these meshes; (iv) a-posteriori estimates on the proposed 3D meshes and extensions to other 3D PDEs with singularities. Intensive parallel computing will play an indispensable role in the research.
We develop new, paradigm-shifting ideas in numerical analysis, by actively incorporating methods from other areas of mathematics, such as geometry and functional analysis. Our long-term goal is to provide cutting-edge numerical techniques for computations across scientific disciplines.