TABLE OF CONTENTS A clustered [eigenvalue] spectrum often translates in rapid convergence of GMRES.
Michele Benzi and Gene H. Golub. SIAM Journal on Matrix Analysis and Applications, Vol. 24 (2003)
The coupled matrix problems represent a vast class of scientific problems arising from discretization of either systems of PDE s or coupled PDE s and integral equations, among other applications such as the Karush Kuhn Tucker (KKT) matrices from nonlinear programming [273,43]. The reader may be aware of the fact that many coupled (nonlinear) systems may be solved by Uzawa type algorithms [92,153,144,115], i.e. all equations are artificially decoupled and solved in turns. A famous example of this strategy is the SIMPLE algorithm widely used in computational fluid dynamics along with finite volume discretization [473,334]. While there is much to do in designing better and more robust preconditioners for a single system such as (1.1), one major challenge in future research will be to solve the coupled problems many of which have only been tackled recently.
This chapter will first review the recent development on a general coupled system and then discuss some specific coupled problems. The latter samples come from a large range of challenging problems including elasticity, particle physics and electromagnetism. We shall discuss the following.
Section 12.1 Generalized saddle point problems
Section 12.2 The Oseen and Stokes saddle point problems
Section 12.3 The mixed finite element method
Section 12.4 Coupled systems from fluid structure interaction
Section 12.5 Elasto-hydrodynamic lubrication modelling
Section 12.6 Discussion of software and a supplied Mfile
