Preconditioning constrained systems

11 November 1999
Dr Andy Wathen
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.
  • Computational Mathematics and Applications Seminar