[Nek5000-users] Spectral coefficients for velocity

[nek5000-users at lists.mcs.anl.gov]

Hi Andrey,

I don't know if you got an answer yet but here are my suggestions.

If you are asking about the theory behind spectral transforms, I  
suggest that you look at appendix B in the book "High-Order Methods  
for Incompressible Fluid Flow" by Fischer, Deville and Mund. In  
particular, Section B.3.1 gives the formula for the spectral  
coefficients uh_k

uh_k = 1/gamma_k \int_-1^1 w(x) u(x) p_k(x) dx

where gamma_k=||p_k||^2 is the discrete norm and w(x) is the weight  
function, both of them associated to polynomial p_k.

If you want to compute these coefficients in Nek5000, then you can  
look at the routine "local_err_est" in the file navier5.f. There, one  
goes to the Legendre space by using

call tensr3(uh,nx,u,nx,Lj,Ljt,Ljt,w)

where uh is the array of spectral elements and the operators Lj and  
Ljt are computed in the routine "build_legend_transform" (also in  
navier5.f) from the GLL points.

I hope that helps.

Best regards,
Nicolas Offermans

Quoting nek5000-users at lists.mcs.anl.gov:

> Hi Neks,
>
> Priviously, the parameters p101 and p103 have been discussed:
>
> "In each direction, on each element, the solution is represented as a
> polynomial of degree N in the reference element (i.e., on the interval
> [-1,1]).
>
> For any Nth-order polynomial there is a modal expansion of the form:
>
>    u(x) = \sum_k=0^N   uh_k phi_k(x)
>
> where   phi_0 = 1, phi_1 = x, and phi_k(x)=L_k(x) - L_{k-1} (x) for k > 1,
> where L_k(x)=kth Legendre polynomial."
>
> How can I obtain these coefficients "uh_k" on the several elements?
>
> Thanks in advance for any suggestion,
>
> Andrey