We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$, induced by the $L_2$~norm, that is $\|f-g\|_2=\sqrt{\int_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious na\"ive algorithm requires $\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\O(n\log^4 n\log\log n)$ arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.