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In peer reviewed numerical papers, the order of accuracy of finite difference and finite volume for PDEs is computed in multiple norms, usually $l_1$, $l_2$ and $l_{\infty}$, and other times $l_{\infty}$ is used with one of the other two. I guess that the reason for $l_{\infty}$ to be used is that you wanna check that the max error goes to zero with the right order. My questions are:

  1. Why does convergence with a certain order in one of the three norms not guarantee convergence with the same order in the other norms?

  2. What is the added information of using $l_2$ beyond what you get from $l_1$?

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3 Answers 3

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In finite dimensional spaces (say, in $\mathbb R^n$), all norms are equivalent and as a consequence, if something converges with a specific rate in one norm, it also converges with the same rate in all other norms. But that is not the case in infinite dimensional spaces (where you are looking for the solution of partial differential equations), and consequently it matters what norm you report convergence in.

As for your second question of why to choose $L_2$ or $L_1$ and not the other: This is often motivated by considering what physical quantities are interesting in an application. For example, for elliptic problems, one often cares about the energy, and so using a quadratic norm makes sense (because the energy is often the integral of the square or something). In hyperbolic equations, the total mass/energy/momentum is often interesting, and these are all integrals of some quantity itself -- so the $L_1$ norm is appropriate to measure errors.

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Both your questions have been addressed in the answer of @WolfgangBangerth. Here is an applied example of the behaviour of the different norms in the context of polynomial approximation (which is for the OP comparable to the solution of ODEs):

enter image description here

The picture is from here and shows the error of the approximation of the function $\sqrt{|x-3|}$ by a polynomial of degree 20, once in terms of the optimal polynomial (pink), as well as in terms of the Chebyshev expansion (blue).

It shows the following points:

  • The optimal polynomial (pink) is better in terms of the $L_\infty$ norm (in fact this is by definition)
  • The Chebyshev polynomial expansion is way better in the $L_2$ norm (and is, as Trefethen points out, probably considered "more optimal" by most users)

For both approximations, stating only one norm would probably be misleading.

Moreover, this example shows that in computational science we don't care that much about the equivalence of norms, but rather on the quantitative behaviour of the numerical approximation. This holds for approximation theory as well a for the solution of PDEs.

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There are horses for courses. Consider the solution to a hyperbolic differential equation, using a scheme that "captures" the discontinuities. Except under very special conditions, the solution on all grids will have an O(1) error close to the discontinuities and there will be no convergence in $L_\infty$. The proper norm (in the sense of most revealing) is $L_1$. The infinity norm can be revealing in the process of debugging (if there are a few large errors where are they?)

The chief advantage of $L_2$ is that it leads to nice analysis. Minimizing in $L_2$ leads to linear equations.

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