[Nek5000-users] Projection of linearized equations onto reduced basis

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

Hi Neks,

I would like to project the linearized NS equations onto a reduced basis 
formed by some POD modes (velocity vectors).
We know that the perturbation equations can be read as du(t)/dx = A 
u(t), where

// A = ( -U.grad(u) - u.grad(U) - grad(p) + 1/Re Lap(u)//

To obtain the projected system, I need to compute the matrix Ar_i,j =  
<U_i, AU_j> where U_j is the j mode (a vector containing the velocity  
field).
The step that  I  can't clearly see is the matrix-vector product A U_j.  
Do  I  have to explicitily compute the matrix A  and then obtain the 
product? or is there a direct  way to  do  it  without this matrix 
(using the time-stepper)?

Thanks in advance!
Sincerely,
Juan--