Relative distance between two scalar fields. Application to mathematical modelling approximation

Inspired in the relative error between two quantities, we define a functional Δ_D that operates on two non‐negative scalar fields, which are integrable in a given open connected set D. We prove that Δ_D defines a metric, but not an ultrametric nor a translation invariant metric. We show how to apply Δ_D to evaluate the performance of analytical approximations of PDEs. The principal advantage of using Δ_D instead of other distances given in the literature is that the value given by Δ_D has a very easy interpretation: a value close to 0 means relatively near, but a value close to 1 means relatively infinitely far. Copyright © 2013 John Wiley & Sons, Ltd.