When can we say that a numerical scheme converge and when does it matter Suppose we have a pde and we want to compute the solution to solve some engineer problem.
I come up with a nice numerical scheme, and i'm happy with that
Now, because I'm a mathematician, i try to have a proof that the scheme converge and at which speed.
Let's say that I come up with a result like this
$$ \| u_h - u \|_{L^2} < Ch\|u_0\|_{H^1},$$
so that if my initial condition is in $H^1$, my solution will converge in $L^2$ norm, nice !
Now imagine I'm able to prove that my solution does not converge in $H^1$. Can I really say that my numerical scheme converge ?
Of course, from a mathematical point of view, the answer is obvious : yes and no, it depends of what notion of convergence you have. But from the point of view of someone who just wants to simulation something, is it good enough ? And if yes, why ?

It really depends on the application. If your application requires approximating $u$ well, then maybe $L^2$ convergence but not $H^1$ is fine. If your application requires approximating the gradient $\nabla u$, then you're in trouble.
For example, in optimal control, one solves a PDE (the Hamilton-Jacobi-Bellman equation) to get the value function $u$, and then the optimal control is a function of the gradient $\nabla u$. In this application convergence of the gradients is very important and one would hope for even better than $H^1$ convergence (say, $C^1$ convergence, but this is often not possible; this is another story...).
In other applications it may not even be possible to obtain $H^1$ convergence. Consider scalar conservation laws, where solutions are discontinuous. You may get $L^2$ convergence, but the solutions do not live in $H^1$ so you can't hope for $H^1$ convergence anyway.
