Which norm should I use?
 

Hi guys,

I have a question on the convergence test.

I am verifying a second-order projection scheme. Benchmark tests show that it indeed has a second-order convergence if the error is measured in L1 norm. However, this is not true if it is measured in L2 or L_infinite norm.

I have seen people conducting grid refinement analysis using different norms. My question is which norm really reflects the accuracy and how?

Thanks in kindly.

Tony

Re: Which norm should I use?
 

1) See which norm with which the accuracy is defined.

This is important because the problem may be convergence under a norm while it may not be convergence under another norm. E.g, any shock-capturing scheme is not convergence under L_infinite norm.

2) For a smooth NSe problem, the convergence rate with respect to difference norms are roughtly same. You may check you program. Usually, people list L_2 norm and L-infty norm since the former is associated with the energy, while the latter one signifies the maximum point-wise difference.

-Paul

Re: Which norm should I use?
 

Hi Paul,

Thanks kindly. I have figured it out.

The test I have done is the cavity-driven flow, where the singularities appear at two of the four corners and cause deterioration of the L-infinite norm. I am wondering is it possible to make the L-infinite norm also second-order converge for the cavity flow problem?

You are right that for smooth problems, all the norms should roughly same, but may not necessary in the same convergence rate. L2 norm is between the L1 and L-infinite norms, right?.

Thanks again.

Tony