A new analysis of block preconditioners for saddle point problems

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur com-plement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular precon-ditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these preconditioners based on symmetric block Gauss–Seidel-type itera-tions. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue dis-tributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow one to localize accurately both real and nonreal eigenvalues. The relativ