When I read books on finite differences they often end up using discrete $L^2$ norm for estimating the error as it naturally arises from weak formulation. I was wondering if people do that in Sobolev norm and when it is useful. I have not seen any at least in finite-differences book.
To be more specific look at the $$Au=f,$$ where assume $A_h$ is some approximation for $A$ and $U$ is the numerical solution for the system. Then if we plug the actual function $u$ into $A_hU=f$ and substruct we have $$A_h(u-U)=\tau$$ for $\tau$ being a local error. Thus I have an error equation $$e=A_h^{-1}\tau$$ What are the problems I am facing If I use discrete Sobolev norm? What would that be then, it should involve derivatives estimates, but can I do one for the local truncation error?
Thanks!