Interval arithmetic for finite difference error bounds

It seems that interval arithmetic can be used to quantify floating point truncation error in computational calculations. Does anyone know if it is possible to use interval arithmetic in a finite difference scheme, to automatically quantify the error of the scheme?

For example, using interval arithmetic, given some complicated f(x), and given a finite precision interval which contains the infinite precision value of x, I can get an exact representation (in finite precision) of an interval which contains f(x). This will allow me to know the impact of the truncation/rounding errors that have occurred during the calculation.

Now, if f(N,x) is an N-point approximation to f(x), can I somehow use interval arithmetic to determine a good interval [f1, f2] which will contain f(x) when computed using the approximation f(N,x)?

In particular, I'm wondering if I can get an error bound of f(x) when it is the solution to a PDE which is solved using a finite difference scheme.

While I have seen this alluded to, I have not seen a good description or reference.