Weak Galerkin methods for time-dependent Maxwell's equations

This paper adapts the weak Galerkin (WG) finite element scheme to Maxwell's equations in the time domain. Developed by Wang and Ye in 2011, the WG scheme is a discontinuous Galerkin-like method that relies on a new notion of the discrete weak derivative. The WG method as applied to Maxwell's equations uses a discrete weak curl and an extra stabilization term to enforce a weak continuity of the solution across elements. Semi-discrete and fully-discrete schemes are developed and are shown to be unconditionally stable. An optimal order of convergence is proven in an appropriate energy norm and confirmed by numerical results. For clarity, the theoretical analysis is carried out for 3-D case. Some details showing how to implement this WG scheme in 2-D are provided and numerical results are given in 2-D.