In general, our methodology is based on eliminating the singularity of the (1,1) block by various means. One straightforward approach is based on estimating a small number of components of the solution vector by using the Lanczos/Stieltjes procedure, and then substituting the computed values into the original system. As a result, the new linear system is smaller in size. Iterative refinement can be applied to improve the accuracy.
In addition, we consider the augmented Lagrangian procedure, which is related to penalty methods. We will provide some observations regarding the condition number, the spectrum, and a sensible choice of the parameters involved. The analysis demonstrates how the case of a singular (1,1) block is different from other cases. We shall also present some results regarding the spectrum of the associated Schur complement. Some numerical examples that illustrate the efficacy of our methods will be presented throughout the talk.