Combination Preconditioning and the Bramble-Pasciak+ Preconditioner

It is widely appreciated that the iterative solution of linear systems of equations with large sparse matrices is much easier when the matrix is symmetric. It is equally advantageous to employ symmetric iterative methods when a nonsymmetric matrix is self-adjoint in a nonstandard inner product. Here, general conditions for such self-adjointness are considered. A number of known examples for saddle point systems are surveyed and combined to make new combination preconditioners which are self-adjoint in different inner products. In particular, a new method related to the Bramble-Pasciak CG method is introduced and it is shown that a combination of the two outperforms the widely used classical method on a number of examples. Furthermore, we combine Bramble and Pasciak's method with a recently introduced method by Schoberl and Zulehner. The result gives a new preconditioner and inner product that can outperform the original methods of Bramble-Pasciak and Schoberl-Zulehner.