The general importance of preconditioning in combination with an
appropriate iterative technique for solving large scale linear(ised)
systems is widely appreciated. For definite problems (where the
eigenvalues lie in a half-plane) there are a number of preconditioning
techniques with a range of applicability, though there remain many
difficult problems. For indefinite systems (where there are eigenvalues
in both half-planes), techniques are generally not so well developed.
Constraints arise in many physical and mathematical problems and
invariably give rise to indefinite linear(ised) systems: the incompressible
Navier-Stokes equations describe conservation of momentum in the
presence of viscous dissipation subject to the constraint of
conservation of mass, for transmission problems the solution on an
interior domain is often solved subject to a boundary integral which
imposes the exterior field, in optimisation the appearance of
constraints is ubiquitous...
\\
\\
We will describe two approaches to preconditioning such constrained
systems and will present analysis and numerical results for each. In
particular, we will describe the applicability of these techniques to
approximations of incompressible Navier-Stokes problems using mixed
finite element approximation.