题目：Pointwise Divergence-Free Spectral and Spectral Element Methods for Maxwell Equations on Spherical Domains

报告人：李会元 研究员 （中国科学院）

地点：致远楼108室

时间：2024年10月18日 16:35-17:35

摘要：In this talk, we first construct, on the basis of Hodge-Holmholtz decomposition theory, a complete system of curl-orthogonal vectorial polynomials exactly satisfying the divergence-free constraints on the unit ball.  Then we propose a pointwise divergence-free novel spectral approximation scheme for the double curl Maxwell equation.  Implementation is discribed in brief and numerical analysis is conducted in the sequel.

Next, we extend our idea to propose a pointwise divergence-free novel spectral element method for the 3D mean-field dynamo equation on a spherical domain with a core zone together with the inner and outer spherical shell zones. Besides, a second order stiffly stable generalized BDF2 discretization is designed in time. At the same time, efficient algorithms for the fully discretized scheme is implemented with the help of the spherical harmonic transformation at each time step. Besides, we analysis the stability of the fully discretization scheme, then prove rigorously the optimal error estimate of the fully discrete scheme. Numerical experiments are finally carried out to validate our algorithms and main theory.

李会元，中国科学院软件研究所研究员，博士生导师，并行软件与计算实验室常务副主任。主要从偏微分方程谱方法、高性能软件与算法等研究工作，研究兴趣包括偏微分方程特征值问题的谱与谱元方法、国产异构平台上高性能数值并行软件的研究与性能优化。现任中国数学会理事、CISAM谱方法及其应用专业委员会（筹）秘书，曾任中国系统仿真协会青年工作委员会委员、北京计算数学会理事。主持多项国家自然科学基金项目、参与多项国家自然科学基金委重点项目，承担多项国家重点研发计划高性能计算专项课题，在谱与谱元方法，非传统傅里叶变换，以及高性能计算机基准测试、数学库研制、数值应用软件性能优化上取得了研究成果。

欢迎各位参加！