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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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This may be a good one for the new scicomp.se private beta. If you think so too, I can invite you. –  qubyte Dec 4 '11 at 15:52
    
Thanks Mark, but since this is already posted here, I'd leave it. Certainly the next question I have will go there. –  user566 Dec 5 '11 at 17:55
    
No problem. It just went into open beta, so if you want to ask a question in the future then there's no need for invites. –  qubyte Dec 7 '11 at 13:44
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2 Answers

up vote 5 down vote accepted

Two collegues of mine wrote a monograph

Faragó, Karátson: Numerical solution of nonlinear elliptic problems

where conditioning and preconditioning are an issue. This may be some help though...

If you want it more elementary, then

Larsson, Thomee

is the best.

Do you have a special equation in mind? That would help.

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Thanks, this is more or less what I had in mind. –  user566 Dec 5 '11 at 18:03
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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.

Math WS547 Numerical Analysis of Elliptic PDE's (3 cr) WSU Math 546

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.

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Thanks for your help. –  user566 Dec 5 '11 at 18:02
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