Parameter-free H(div) preconditioning for a mixed finite element formulation of diffusion problems

Mixed finite element formulations of generalised diffusion problems yield linear systems with ill-conditioned, symmetric and indefinite coefficient matrices. Preconditioners with optimal work complexity that do not rely on artificial parameters are essential. We implement lowest order Raviart-Thomas elements and analyse practical issues associated with so-called 'H(div) preconditioning'. Properties of the exact scheme are discussed in Powell & Silvester (2003, SIAM J. Matrix Anal. Appl., 25, 718-738). We extend the discussion, here, to practical implementation, the components of which are any available multilevel solver for a weighted H(div) operator and a pressure mass matrix. A new bound is established for the eigenvalue spectrum of the preconditioned system matrix and extensive numerical results are presented. © The author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.