Appropiate models of numerical computation Hello,
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.
 A: The principal motivation for the Blum Shub Smale model of computability was precisely the kind of concern you raise in your question. In particular, the BSS machines provide a numerical model of computation using a random access machine concept, where the registers hold full-precision real numbers. The dynamicists had wanted a theoretical model of computability that would untangle the discrete computational effects, such as round-off error, from the computational analysis of numerical algorithms involving continuous quantities. They wanted to provide a formal setting in which to analyze issues such as stability and convergence of algorithms in a more continuous setting, where quantities would be represented with perfect precision, and the typical discrete computational issues would loom less large. The BSS model is provably different from the model of computable analysis.
A: There are mainly three approaches to deal with computational complexity of continuous problems.
1. Information Based Complexity (analytical complexity). It is a very general framework that describes the complexity of a problem in terms of the number of 'operations' (specified by the problem itself) needed to solve it on condition that we are given only rough information about the initial problem. This theory is mostly about the 'real' complexity --- which is independent of any particular model of computation --- it gives lower bounds on all possible models. See 'Information-Based Complexity' by Traub, Wasilowski and Woźniakowski for more details.
2. Blum–Shub–Smale machine and that like (algebraic complexity). The idea comes from the good old days when people believed that it was possible to build analog computers that are more powerful than turing machines. I was to say that the idea of stroing infinite information in a single cell, and comparing two such cells in a finite time is a bit crazy from both practical and theoretical point of view, but I guess it is prudent to refrain from making such comments. These models are inconsistent with physical laws, so it should not be strange that they allow some 'dirty hacks' (for example there are uncomputable problems easily solvable on Blum–Shub–Smale machine; under some definitions, one may show that both $P/Poly$ and $NP$ are solvable in polynomial time).
3. Turing machines (discrete complexity). If you really want to solve a problem on a computer, then you really have to transform it into a discrete one (either symbolic or numeric). But then, there is nothing left but the classic complexity :-)
David, I do agree with your reasoning, and (so) with your conclusion. However, notice that it is always easy to falsify statements such as my remarks. Simply, the reality is so complex, that in every such proclamation there must be much more things that we have to ignore, than we are able to take into consideration. The crucial point here is that BSS allows us to perform dirty tricks and reach something paradoxical, which otherwise we would have not accepted; and that the extra power does not give us anything besides these paradoxes. Every algorithm for BSS either: has its counterpart in the standard model, or is ‘unrealizable’ (we may have an infinite precision, and infinite memory, but when we cut these infinities at any stage, we get a Turing machine). Put it differently, according to our current knowledge every ‘sensible’ ‘realization’ of a BSS has to factor through a Turing machine.
A: Caveat emptor: answer written by a non-expert.
I think Algebraic Complexity Theory in general provides a useful model for dealing with computations involving real numbers (or other fields). The BSS model seems to be one successful model in algebraic complexity. 
A question of my own is: isn't interval arithmetic suitable for the kind of analysis that one might wish to perform for numerical algorithms? 
Another question is: What about Non-standard analysis?
