Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\mathbf{L}}^2$-norm error estimates

We present a novel parameter-free discontinuous Galerkin (dG) finite element method (FEM) for the time-dependent Maxwell system formulated as a saddle point problem. We establish the stability of the proposed semi-discrete problem and derive optimal error estimates in energy and ${\mathbf{L}}^2$ norms for the electric field variable, as well as in $L^2$ norm for the potential function. To the best of our knowledge, this work provides the first optimal ${\mathbf{L}}^2$-norm error analysis for the second-order time-dependent saddle point Maxwell equations using any variants of FEMs. Additionally, we propose several complete discrete time-integrators and verify the optimal convergence results through examples in both 2D and 3D setups.