Auxiliary Space Preconditioning for Edge Elements

We present a general approach to preconditioning large sparse linear systems of equations arising from conforming finite-element discretizations of -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 second-order elliptic problems have to be available, which are provided, for example, by standard algebraic multigrid codes. In a sense, the method described in this paper enables us to construct optimal algebraic preconditioners for discrete -equations.