Apologies for the stupid question, it's been a while and I'm finding the literature confusing. I've commonly seen implemented, finite difference methods where the american option price is set to be the maximum of the solved value at a particular node, and the immediate exercise value at that point. Alternatively, I've seen a fair number of SOR implementations with an iterative solution.Is the former correct or merely a convenient "approximation"? Is there a definitive paper on this? It seems like there's an awful lot proved in terms of getting to the method or supporting methods but difficult to tell which the relevant results are in terms of implementing the method.

Taking the maximum of the solved value and exercise value at each time step does give an approximation.It should converge to the right value when the space-time discretization is refined.The projected SOR iteration and many other methods give a more accurate approximation that the aboveprojection at each time step.

Thanks. Are there any good references dealing with these results?

A complete treatment is John Crank "Free and Moving Boundary Value Problems". All methods in QF tend to be based on it. He discusses about 5 different methods. This book is really essential if you are serious about PDE. G.H. Meyer has a number of articles on this.