We present a general approach to preconditioning large sparse linear systems of equations arising from conforming finite element discretizations of H(curl, Ω)-elliptic variational problems. Like geometric multigrid, the methods are asymptotically optimal in the sense that their performance does not deteriorate on arbitrarily fine meshes. Unlike geometric multigrid, no hierarchy of nested meshes is required, only fast solvers for discrete Poisson problems have to be available, which are provided, e.g., by standard algebraic multigrid codes. In a sense, the method described in this paper paves the way for constructing optimal algebraic preconditioners for discrete curl curl-equations.
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