The Landau-Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint |m| = 1. In this talk, I will present an overview of recent development on numerical methods and analysis for the Landau-Lifshitz type equation. A crucial issue in designing efficient numerical schemes for this equation is to preserve this constraint in the discrete level. A simple and frequently-used one is the projection method which projects the numerical solution onto a unit sphere at each time step. Due to the simplicity of the sphere-projection approach, the method has been extensively used in various applications, including for energy-conserving or symplectic system and the evolution on a manifold. However, no rigorous error estimate is available up to now. Classical energy approach fails to be applied directly in the analysis of the projection method since both projected and unprojected solutions are involved in the discrete system. We shall present our recent works on optimal error analysis of linearized finite difference and finite element methods for the Landau-Lifshitz equation. The analysis is based on a more precise estimate of the difference between the errors of projected and unprojected solutions. Some numerical experiments are provided to confirm our theoretical results.

孙伟伟教授，西北工业大学学士，西安交通大学硕士，加拿大温莎大学博士，专业为应用数学。知名计算数学专家，曾担任香港城市大学教授，2020 年 1 月加入 UIC。主要的研究方向是科学计算与数学建模，包括偏微分方程数值解及理论分析、数值线性代数、纺织材料中热和水汽运动数学建模等， 近几年针对非线性抛物问题提出了一套新的框架性的分析方法—无条件误差估计。孙伟伟教授担任以下期刊的编委：《 International Journal of Numerical Analysis and Modeling》和 《 Numerical Mathematics: Theory, Methods and Applications》，发表科研论文上百篇，其中在SIAM系列期刊上合作发表论文30余篇。