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In most of the books on numerical methods and finite difference methods the error is measured in discrete $L^2$ norm. I was wondering if people do the in Sobolev norm. I have never see that done and I want to know why no one uses that.

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?

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For one thing, it's a question of what norm measures how "accurate" the solution is. Which of the two error terms would you rather have: $0.1\sin(x)$ or $0.0001\sin(10000x)$? The first is smaller in the Sobolev norm, the second is smaller in the $L^2$ norm.

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but if I choose to use Sobolev norm because I have control on not only how function behaves but also on the derivative, which might be of advantage, I have to estimate then $||A_h^{-1}||_{H^1}$ and $||\tau||_{H^1}$ is that right? I am not sure such a norm for matrix exist.... – Medan Jun 19 '12 at 13:33
@Medan There is at least reasonable discrete Sobolev norm on vectors, namely $\sum_{k} |x_{k}-x_{k+1}|^2+(n^{-1}\sum_k x_k)^2$. I threw in the average of $x$ to make it a norm rather than a seminorm. For any vector norm $\|x\|$ there is a corresponding matrix norm, namely $\|A\|=\sup_{\|x\|\le 1}\|Ax\|$. This definition is not great from the computational viewpoint, though. Unfortunately, I don't know numerical analysis, so everything I say is just a layman's guess. Looking up "discrete Sobolev norm" is more likely to be helpful. – user31373 Jun 19 '12 at 14:43

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