Where to get started with Numerical Solutions to PDEs?

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I have established existence and uniqueness of solutions to a nonlinear elliptic system of PDE's and would like to see how the solutions look like numerically. I have some programming background on MatLab and C++. I also understand basic ODE numerical schemes. But I am not so sure, if I need to begin by first reading books on Numerical Analysis or is there a more direct way to learning about numerical solutions to my particular problem?

Thanks in advance for your discussion.
 
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I'm not an expert on numerical methods for PDE's but since you haven't received other replies, I'll suggest that you look at material on "Finite Element" techiques if they apply to your PDE's. The Finite Element books I have glanced at (such as the Shaum's book) are very concrete. You can probably find books that do Finite Element analysis in particular programming languages. I notice the wording in one of the Shaum's books implies that older books on Finite Elements are obsolete because of modern developments. I don't understand any details about that, but it wouldn't hurt to get a modern book.
 
Google 'nonlinear PDE'.
Let your fingers do the walking. There are several references to the numerical solution to nonlinear PDEs on the first page alone.
 

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