How does one calculate the critical stock price in the finite difference methods for calculating American optionprices ?

You need to use the SOR or some other method suitable for free-boundary problems. If you employ the same method as for European options you will get a mistake Anton

There is a paper from P. carr, who adresses the same issue. [american options with finite difference for jump diffusion]. He gives the whole algorithm, but does not consider the determination of the optimal exercise boundary.This method is a kind of-layer-by-layer method: so you can proceed to layer n, if you have done layer n-1.I would guess the following: In my point this is an unknown parameter in the system matrix : you should write up the smooth pasting condition and you should find somehow a homogenous linear matrix equation: then the det A = 0 may give the opt. ex. boundary.

when i did this, i followed the little red book by Wilmott, Howison and Dewynne

Exponentially fiited schemes, for example do the job. At each time level take constraints into account.See the Software forum for a number of comments on this topic.