Large linear systems in saddle point form arise in a wide variety of scientific computing problems. These include mixed formulations of second-order and fourth-order elliptic problems, incompressible fluid flow problems such as the Stokes and Oseen equations, certain formulations of the Maxwell equations, and linear elasticity problems. Often, saddle point systems occur from the use of Lagrange multipliers applied to the minimization of an energy functional subject to a linear equality constraint. The solution of saddle point problems ("KKT systems") is also of central importance in many approaches to constrained optimization, including the increasingly important field of PDE-constrained optimization.

In the first part of my talk I will discuss properties of saddle point matrices, including spectral properties relevant to the iterative solution of these systems by preconditioned Krylov subspace method. In the second part of the talk I will give an overview of the most effective methods currently available for solving large-scale saddle-point problems, with a focus on augmented Lagrangian-based block preconditioners designed for incompressible flow problems (joint work with Maxim Olshanskii and Zhen Wang).