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An efficient iterative solver is proposed for the linear matrix equation that arises in the analysis of large scattering problems by the frequency-domain finite-element method. A Krylov method is preconditioned by a technique that approximately solves equivalent problems in two auxiliary spaces: a space of scalar functions and a space of piecewise linear, “nodal,” vector functions. On each space the traditional algebraic multigrid method is employed. Further, the “shifted Laplace” idea is used to improve the performance of the solver as the frequency increases. Results are reported for a waveguide cavity filter and three free-space scatterers: a conducting sphere, a metallic frequency selective surface, and a metamaterial lens made of split-ring resonators containing dielectric and metal.