in contrast to the more discrete part of computational mathematics (cryptography, combinatorial computation), numerical mathematics seems to ignore typical questions of theoretical computer science -- what does 'algorithm' or 'computation' mean, what is the model of computation.
This is far from fallacious. For example, a finite element theorist mostly investigates only approximation schemes and convergence rates, which in principle do not demand any computation at all. Algorithms are typically neat and short, and any exceptions to these are not rarely just combinatorial insertions for mesh management. The other end of the spectrum compromises very technical numerical mathematics. - In either case, the 'deep-down' part is largely abandoned as soon as possible, because it is largely irrelevant. Just like no cryptographer enjoys talking about Turing machines.
Has there been a rigourous treatment of a numerical model, a justification why (or for whom) a certain model might be appropiate?
I am aware of computable analysis and numerical mathematicians who participate in this field. But I am not aware of a numerical model like, say, a numerical random access machine. I even suppose there different models appropiate for researchers in fundamental numerical algoriths or a FEM reasearcher, depending on which level of detail is needed.