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.

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