Approximation algorithms for problems in P One usually thinks about approximating solutions (with guarantees) to NP-hard problems. Is there any research going on in approximating problems already known to be in P? This might be a good idea for several reasons. Off the top of my head, an approximation algorithm may run with a much lower complexity (or even a much smaller constant), might use less space or might be much better parallelizable.

Also, schemes that provide time/accuracy tradeoffs (FPTAS's and PTAS's) might be very attractive for problems in P with lower bounds that are unacceptable on large inputs.

Three questions: is there anything that I'm missing that makes this obviously a bad idea? Is there research going on in developing a theory of these algorithms? If not, at least, is anyone familiar with individual examples of such algorithms?
As Jukka points out, computational geometry is a rich source of problems that can be solved in polynomial time, but we wish to get fast approximations. The classic "ideal" result is an "LTAS" (linear time approximation scheme) whose running time would be of the form $O(n + \text{poly}(1/\epsilon))$ - usually these are obtained by extracting a constant (poly($1/\epsilon$)) sized kernel from the data, and running an expensive algorithm on that kernel, with a guarantee that an exact solution on the kernel is an approximate solution on the entire input. 

There are a number of tricks, reductions and principles, and Sariel Har-Peled's new book is full of these. I don't think there's a rich complexity theory as such.