Analysis of a Galerkin finite element method for the Maxwell–Schrödinger system under temporal gauge

Abstract

Abstract This paper is concerned with the numerical solution of the Maxwell–Schrödinger 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.