The Crank–Nicolson mixed finite-element method for time-dependent Maxwell’s equations under a nonlinear boundary feedback with delay

In this paper, we consider time-dependent Maxwell’s equations under a nonlinear boundary feedback with delay. By employing a mixed finite-element method for spatial discretization and the Crank–Nicolson method for temporal discretization, we construct a fully discrete scheme. The numerical stability and the suboptimal convergence rate of order O(hα -1/2 + t²) are established. Furthermore, a distinct interpolation error estimate for the tangential components of functions on the boundary is derived, which relates the convergence rates to the boundary norms rather than the volume norms. Using this interpolation error estimate, the scheme is shown to achieve the optimal convergence rate of order O(hα + t²), provided that the tangential component of the electric field is sufficiently smooth on the boundary. Finally, numerical examples are given to demonstrate our theoretical results.