Finite element convergence for the Darwin model to Maxwell's equations

Abstract

In three dimensional polyhedral domains with a Lipschitz continuous boundary, we derive the H(curl ; Ω) and H(curl, div ; Ω) variational formulations for the Darwin model of approximation to Maxwell's equations and prove the well-posedness of the variational systems. Then Nedelec's and standard finite element methods are used to solve two kinds of variational problems. Though symmetric bilinear forms in the variational systems fail to define full norms equivalent to the standard norms in the finite element subspaces of H(curl ; Ω) and H(curl, div ; Ω), we can still prove the finite element convergence and obtain the error estimates, without requiring the physical domains to be convex.