I need to understand what is the right expression for the "discrete $H^1$-norm of a function v(which of course need to be in $H^1$).
By definition, $$||v||_{H^1}^2 = ||\nabla v||_{2}^2 + ||v||_2^2 $$
so far I'm still in the continuous case. In practice if I need to compute it I have to do the discrete $L^2$ norm of the gradient and of $v$.
For $v$ I have: $$||v||_2^2 = h \sum_{i=0}^{N} v(x_i)^2$$
But what about the $$||\nabla v||_2^2$$? How can I approximate it?