Saddle point least squares iterative solvers for the time harmonic Maxwell equations

Abstract

We introduce a least squares method for discretizing the time-harmonic Maxwell equations written as a first order system. The method belongs to the recently introduced least squares approach called the Saddle Point Least Squares (SPLS) method (see Bacuta and Qirko 2015, 2017). It is related to two known methods: the Bramble and Pasciak least squares method for solving electromagnetic problems (Bramble and Pasciak, 2004; Bramble et al., 2005) and the minimum residual DPG method of Demkowicz and Gopalakrishnan (2010, 2013). The proposed discretization is based on reduction to a symmetric saddle point formulation on spaces of piecewise polynomial functions. The edge elements are avoided. A discrete test space is chosen first among standard conforming finite element spaces and the discrete trial space is chosen second as the image of the first order Maxwell differential operator on the discrete test space. For the proposed iterative processes, a nodal basis for the trial space is not required and the main inversion at each step is done only on the test space. The method is efficient for both convex and non-convex domains and for a various range of frequencies as shown by the numerical results.