Transition from theory of PDEs to applied analysis and industrial problems and models with PDEs Suppose one wants to transition from the study of certain theoretical aspects of PDEs (say, regularity theory for elliptic operators) to a career in industry solving real-world problems about PDEs and modeling.
What resources are useful in such a transition? 

More specifically, what books offer exensive explanations of concrete complex industrial problems involving PDEs and ways to solve them? 

To be clear: I'm not looking for a textbook that gives some physical motivation of the heat equation on the cylinder. 

I'm rather looking for  books that deal analytically with concrete examples of complicated (possibly messy) problems and PDE models that arise in real industrial settings (and possibly provide MATLAB codes as well).

 A: The best place for answers to your question is to look in the literature. There you will find the most up to date and advanced applications of PDEs to the real world.
That said a strong understanding of the theoretical foundations of PDEs is helpful, just applying a numerical algorithm to a PDE problem never gives a complete picture.
For a solid resource on theory of complex non-linear PDEs I recommend  An Introduction to Nonlinear Partial Differential Equations by J. David Logan, which you should read before going head first into the literature. 
Next if you are looking for a career in PDEs you need to develop a strong understanding of numerics after understanding the theoretical basis of PDEs.
For concrete examples of solving PDEs in the real world see


*

*Modeling, Analysing, and Simulating Problems from Practice

*This review of PDEs in Ecology which while more theoretical has really good references to field experiments.

*PDEs in medicine

*Image Processing
A: It is difficult to be specific without knowing a bit more about you. I suggest starting with the booklist of the Society for Industrial and Applied Mathematics (siam.org). This will be by no means restricted to numerical methods, but it will include them. Good luck!
