Continuous limit and nonlinear functional analysis

I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $Phi$ from an open set (in an infinite dimensional Banach space) of functions to itself, I try to find fixed points of this map $Phi(f)=f$, or equivalently to solve the roots of of $Phi’=Phi-{bf 1}$ (where $bf 1$ is identity functional ${bf 1}(f)=f$) via some Newton method. (in my case $Phi(f)(x) sim int f^n(y) K(x,y) dy$ for some integer $n>1$ and $K$ some integration kernel).

I am really a newbie in this and I have tried several types of discretization schemes, using polynomial interpolants and discretizing the functions $f$ on a grid and using piecewise-linear interpolants.

I am particularly interested in the theory that would tell me why or why not would the second option (with piecewise linear interpolants) converge well or not to the solutions of the continuous equation when I let the size of the grid tend to infinity, i.e. the spacing between two elements in the grid go to zero.

I empirically observe some erratic behaviour, and, while I have some physics intuition of why such limits can go bad sometimes, I would like to do some actual reading on this rather than doing the analysis myself like an amateur at this point, as I think this is some rather standard theory belonging maybe to Approximation theory or something. Google didn’t quite help me on this.

(I originally asked on mathematics.stackexchange but discovered this scicomp and figured it might be a better place to ask).