The Crank–Nicolson Galerkin method and convergence for the time-dependent Maxwell–Dirac system under the Lorentz gauge

Abstract

This paper discusses the numerical algorithm and its convergence for solving the time-dependent Maxwell–Dirac system with the perfect conductive boundary conditions under the Lorentz gauge. An alternating Crank–Nicolson Galerkin finite element method for solving the problem is presented. This algorithm preserves the conservation of the mass and energy of the system. The sharp error estimates for both the solution and the energy are derived. Numerical test studies are then carried out to confirm the theoretical results.