Theoretical question about convergence rate

[Filipe Brandao]

Hi,

I have a quick question regarding the convergence rate of nek5000. I know that the convergence rate should be exponential for the p-refinement (increasing polynomial order while keeping the same number of mesh elements).

But what about the h-refinement (keeping polynomial order the same and increasing the number of mesh elements)? I think I saw somewhere it should be N+1 (N = polynomial order) but not really sure. Most papers I found just show for the p-refinement.

[Fischer, Paul]

Yes - it should be O(h^N) (or perhaps N+1, depending on the norm, but for typical values of N used in Nek the "+1" is largely immaterial).

Hth.

Paul

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From: nek...@googlegroups.com <nek...@googlegroups.com> on behalf of Filipe Brandao <filipe....@gmail.com>
Sent: Monday, February 2, 2026 11:41 AM
To: Nek5000 <nek...@googlegroups.com>
Subject: [nek5000] Theoretical question about convergence rate

 

Hi,

I have a quick question regarding the convergence rate of nek5000. I know that the convergence rate should be exponential for the p-refinement (increasing polynomial order while keeping the same number of mesh elements).

But what about the h-refinement (keeping polynomial order the same and increasing the number of mesh elements)? I think I saw somewhere it should be N+1 (N = polynomial order) but not really sure. Most papers I found just show for the p-refinement.

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[Filipe Brandao]

Hi Paul, thanks for the response.
 
is 'h' the mesh size? So if I reduce the mesh element size by half with a 7th polynomial, it should be (0.5^7)?

[Fischer, Paul]

Exactly.... If you're in the ball of convergence, then you should see the asymptotic behavior that you indicate, provided the overall error is not overwhelmed by something else such as temporal discretization error, etc.

Keep in mind, if you have a solution that looks like u(x)=sin(500x), you are not going to see good convergence (with any code) until the signal is resolved... then you'll start converging and see the O(h^7) convergence.    Before that, you're in the pre-asymptotic stage...

Hth!

Best,
Paul

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From: nek...@googlegroups.com <nek...@googlegroups.com> on behalf of Filipe Brandao <filipe....@gmail.com>

Sent: Monday, February 2, 2026 12:19 PM
To: Nek5000 <nek...@googlegroups.com>
Subject: Re: [nek5000] Theoretical question about convergence rate

 

To view this discussion visit https://groups.google.com/d/msgid/nek5000/6a46f7c7-4fbd-4ee6-89e4-0ba7418fff30n%40googlegroups.com.

[Filipe Brandao]

Hi Paul, thanks again for the response.

Can you send me a reference in the literature where the O(h^N) or even O(N+1) is given? As I mentioned previously, all I ever found was convergence rates for the p-refinement.

[Fischer, Paul]

Hi Filipe,

The results above from  are from Quarteroni, Sacco, Saleri's book  and state that convergence is O(h^l), where l=min(k,s-1), where k is the approximation order (e.g., N) and s is the regularity order..., which might be infinity or it might be finite if u has limited regularity.

Hope this helps!

Paul

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From: nek...@googlegroups.com <nek...@googlegroups.com> on behalf of Filipe Brandao <filipe....@gmail.com>

Sent: Monday, February 2, 2026 1:10 PM

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