My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space for the problems. And there are at least two very interesting applications (from my point of view). The first is the solving of continuum mechanics problems (Finite Element,Ritz methods) and the second is solving of systems of linear equations (Krylov subspace methods).
The question is the following: Is there a theory or some part of analysis that studies projection methods in all their applications? If so, can other methods, like finite volume methods, be built from this starting point?
I studied FEA at university but at the moment, all the discrete approximations are like a set of isolated "tools" that I can use in some particular case. Thanks.