Convention in numerical methods error estimates explained

Question

The common look of any numerical methods (like FEM) is: For a function $u$ and an approximation $u^*$ there exists a $C>0$ usually depending on somthing like the space dimension, mesh width, function degree, etc. such that $$\|u-u^*\|\leq C\|u\|.$$ Why is it that these estimates always depend on $\|u\|$ on the right hand side? Is that, helpful in any way, or just impossible to get rid off?

Answer 1

Bounding the relative error as done here gives more universal error estimates and allows to compare problems on different scales. It tells you that $10u^*$ is as good an approximation of $10u$ as $u^*$ was for $u$.