Numerical solution of saddle point problems

Presentation of the course, introduction to saddle point problems, examples, main assumptions, solvability conditions, algebraic and spectral properties of saddle-point matrices. Overview of solution algorithms, direct vs. iterative methods, Schur complement reduction, null space methods, augmented Lagrangian formulation, factorization of saddle-point matrices, remarks on sparse direct solvers (fill-in, elimination graph, reorderings, minimum degree, band-reducing heuristics). Stationary iterations (Arrow-Hurwicz, Uzawa, inexact and preconditioned variants), Hestenes' Method of Multipliers, introduction to Krylov subspace methods. More on Krylov methods: convergence analysis; flexible, inexact, and preconditioned variants, asymptotic convergence rates for sequences of problems of increasing size, field-of-values analysis. Block preconditioners for saddle point problems (block diagonal and block triangular, constraint preconditioning), spectral analysis, exact vs. inexact, augmented Lagrangian-based. Block diagonal/triangular preconditioners for the Stokes and Navier-Stokes equations; Hermitian-skew Hermitian (HSS) and Modified HSS (MHSS) preconditioning. Dimensional Splitting and Relaxed Dimensional Factorization preconditioning for the Stokes and Navier-Stokes equations. Double/multiple saddle point problems, the coupled Darcy-Stokes problem.