I am looking for an elementary reference regarding issues of stability in numerical analysis of non-linear elliptic PDEs, particularly using the finite difference method (but something more comprehensive, covering more sophisticated methods, is even better). Ideally, something aimed at a physicist or applied math audience would be easiest for me to read, but that is not a strict requirement. To be clear, implementation of the method (using your favourite software or programming language) is not the issue, I am more interested in the mathematics involved in numerical stability.
Two collegues of mine wrote a monograph
where conditioning and preconditioning are an issue. This may be some help though...
If you want it more elementary, then
is the best.
Do you have a special equation in mind? That would help.
Moshe, do you have access to coursework at the University of Idaho? They have a course listing that has your question in the title of the course.
However, they don't seem to describe the course itself beyond that.
I did find this paper entitled: LECTURES on COMPUTATIONAL NUMERICAL ANALYSIS of PARTIAL DIFFERENTIAL EQUATIONS (PDF) that may be more in line with what you are looking for (specifically chapter 2).
As for software, MUDPACK is one that I managed to find.
Hope that helps.