Assume I have a multidimensional grid $G$. I consider two functions $f(x)$ and $g(x)$. I have solved the values for the functions over all grid points $x \in G$. Let me now be interested in some third function $F(f(x),g(x))$. I am looking to evaluate $F$ at a point $y$ that is not a grid point using interpolation.
There are two approaches. First I could find $f(y)$ and $g(y)$ using interpolation and then plug them into $F$. Alternatively I could solve for $F$ over the grid and then interpolate on those values directly.
Which method should yield more accurate results? Is there some discussion on this issue in the literature? Does the answer depend on the interpolation method, e.g. multilinear vs. multidimensional spline interpolation?