Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations

We are interested in the numerical solution of large structured indefinite symmetric linear systems arising in mixed finite element approximations of the magnetostatic problem; in particular, we analyse definite block-diagonal and indefinite symmetric preconditioners. Relating the algebraic characteristics of the resulting preconditioned matrix to the properties of the continuous problem and of its finite element discretization, we show that the preconditioning strategies considered make the Krylov subspace solver used insensitive to the mesh refinement parameter, in terms of the number of iterations. In order to achieve computational efficiency, we also analyse algebraic approximations to the optimal preconditioners, and discuss their performance on real two- and three-dimensional application problems.