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.