Adaptive Finite Element Methods for the Unsteady Maxwell’s Equations

In this paper we present a posteriori error estimates, and stability estimates for the time-dependent Maxwell system of electromagnetics. We use the weak formulation of Lee-Madsen and Monk, for which there is an a priori convergence theorem derived by Monk. We then discretise the problem using a standard Galerkin method, and we show that this method is stable. We derive the Galerkin orthogonality properties, which together with some interpolation properties for the finite element solution, Friedrich div-curl inequality, and the strong stability estimates for the adjoint problem (which we also derive) enable us to prove an a posteriori er-ror estimate in the H1-norm that forms the basis for the adaptive algorithm