This paper is concerned with the numerical solution of the Maxwell–Schrodinger system under the temporal gauge, which describes light–matter interactions. We first propose a semidiscrete finite element scheme for the system and establish stability estimates for the finite element solution. Due to the lack of control over its divergence we cannot get  |$\textbf{H}^{1}$| a priori estimates for the vector potential, making it difficult to obtain error estimates by usual techniques. We apply an exhaustion argument to overcome this difficulty and derive error estimates for the finite element approximation. An energy-conserving time-stepping scheme is proposed to solve the semidiscrete system.

　　Publication: 

　　IMA Journal of Numerical Analysis, 2022, Volume 42, Issue 4, pp: 3609-3631.

　　Author: 

　　Chupeng Ma 

　　Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing, Heidelberg University, 69120 Heidelberg, Germany 

　　Email: machupeng@lsec.cc.ac.cn  

　　Yongwei Zhang 

　　School of Mathematics and Statistics, Zhengzhou University, 450001 Zhengzhou, China 

　　Liqun Cao 

　　LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China and 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049 Beijing, China 

　　Email: clq@lsec.cc.ac.cn