Suppose I have the heat the following one dimensional PDE for the heat equation: $$ \frac{\partial u}{\partial t } = \alpha \frac{\partial^2 u}{\partial t^2 } $$ which I discretized in the spatial domain to give me a Linear Time invariant system of the form (based on some given boundary conditions): $$ \dot{u}(t) = Au(t) + b(t) $$ Note that $A$ is independent of time. My question is: are there any special preconditioners for $A$ obtained as above which make use of the fact that $A$ is obtained from a PDE? In Chapter 5 of [1], the authors say that "Good preconditioning strategies have derived for specific type of matrices, in particular, those arising from discretization of PDEs". But nothing is concrete is mentioned and no reference is given.
Any help would be greatly appreciated. Thanks.
[1] Jorge Nocedal, S. Wright. Numerical Optimization. Springer-Verlag New York, 2006. ISBN: 978-0-387-30303-1. 2nd edition.