Multiple preconditioners for saddle-point and other problems

Temple University, Philadelphia PA

Investigators

Abstract

There are many problems in science and engineering where one deals with two completely different phenomena, but these two phenomena influence one another (or one influences the other). One example of this situation is fluid flow along two different kinds of media. One flow is free, as in water in a river, and the other is constrained by some porous media, as the fluid seeping through below the river bed. When one tries to describe these physical phenomena, one needs to use very different equations for each of the two, in addition to taking into account what happens in their interphase. In this project, the PI shall study the solution of systems of equations of the kind just described. The PI shall combine solution techniques developed for each of the two phenomena, so that the resulting combination gives us an efficient computational tool. Among the phenomena to which the PI wants to contribute better modeling and solution techniques is the so-called coupled Stokes-Darcy flow, with the differential equations discretized by finite elements, and in particular by discontinuous Galerkin in the Darcy region. The PI proposes to use new exact and inexact constraint-type (indefinite) preconditioners. The PI will study spectral (and field-of-values) equivalence bounds, which should provide mesh independence bounds on the GMRES convergence for these problems of saddle-point type. Multipreconditioning is another tool that the PI wants to develop for the solution of saddle-point and other problems. In the case of Multipreconditioned GMRES, the PI proposes to provide convergence analysis through the use of multivariate polynomials. In this approach, at each iteration, the method computes in an implicit manner the optimal combination of two preconditioners. For certain saddle-point problems this dynamical combination preconditioning is expected to be more efficient than either of the two original preconditioners.