Error norm

An error norm measures a numerical error vector or error function in a specified norm. For nodal errors $e_j=U_j-u(x_j)$ on a one-dimensional uniform grid of spacing $h$, common choices areThe first controls the largest nodal error. The second approximates a continuum L2 norm on a fixed domain. For nonuniform grids, quadrature or finite-element mass weights replace the common factor $h$.

Mesh weighting matters in a refinement study. If $e_j=\epsilon$ on $N$ cells of a unit interval with $h=1/N$, then $\Vert e{\Vert }_{2,h}=|\epsilon |$, whereas the unweighted Euclidean norm is $\sqrt{N}|\epsilon |$. The latter grows solely because the number of entries grows. Compare the same norm on the same physical domain and, for an evolution problem, at the same physical time.