Efficient finite element solvers for the Maxwell equations in the frequency domain Original Research Article

The present paper shows that certain instabilities encountered with Nedelec-type finite element implementations of the vector wave equation can be eliminated by a family of Lagrange multiplier methods. The considered approaches can be interpreted as coupled vector and scalar potential methods, including an ungauged formulation. We advocate the latter form for use with iterative solvers. We discuss an inexpensive high-frequency variant of the method and show how hierarchical finite element bases can be utilized to derive efficient, partially gauged formulations of higher order. Based on the multigrid idea, a Schwartz-type solver is constructed which can overcome a major insufficiency of conventional preconditioners.