Acquired Clustering Properties and Solution of Certain Saddle Point Systems

Many mathematical models involve flow equations characterized by nonconstant viscosity, and a Stokes-type problem with variable viscosity coefficient arises. Appropriate block diagonal preconditioners for the resulting algebraic saddle point linear system produce well-clustered spectra, except for a few interior isolated eigenvalues which may tend to approach zero. These outliers affect the convergence of Krylov subspace system solvers, causing a possibly long stagnation phase. In this paper we characterize the influence of the spectral properties of the preconditioner on the final spectrum of the saddle point matrix by providing accurate spectral intervals depending on the involved operators. Moreover, we suggest that the stagnation phase may be completely eliminated by means of an augmentation procedure, where approximate spectral eigenspace information can be injected. We show that the modifications to the original code are minimal and can be easily implemented. Numerical experiments confirm our findings.