A least-squares approximation method for the time-harmonic Maxwell equations

In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negative-norm least-squares algorithms for electromagnetic problems which have already been applied to the electrostatic and magnetostatic problems as well as the Maxwell eigenvalue problem (see [4,5]). The scheme is based on a natural weak variational formulation and does not employ potentials or 'gauge conditions'. The discretization involves only simple, piecewise polynomial, finite element spaces, avoiding the use of the complicated Nédélec elements. An interesting feature of this approach is that it leads to simultaneous approximation of the magnetic and electric fields, in contrast to other methods where one of the unknowns is eliminated and is later computed by differentiation. More importantly, the resulting discrete linear system is well-conditioned, symmetric and positive definite. We demonstrate that the overall numerical algorithm can be efficiently implemented and has an optimal convergence rate, even for problems with low regularity.