Null-space preconditioners for saddle point systems

The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive-definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization, and compare their performance with the equivalent Schur-complement based preconditioners. We also describe how to apply the non-symmetric preconditioners proposed using the conjugate gradient method (CG) with a non-standard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications.