Hi, everybody, An optimzation problem on a turbulent flow leads Navier Stokes equations to a new system (by frechet differential and adjoint entity,... but it is not the matter) So, I have especially to resolve this equation ( numerically obviously ).

P_ i + P_ j ( dv_ i / dx_ j ) - v_ j ( P_ i / dx_ j ) - L_ {jj} P_ i = 0

Where v_i and dv_i/dx_j are known. L_{jj} is the Laplacian operator. And _i is the Einstein notation. We have to resolve P_i by an implicit method. AND I REALLY DON'T KNOW HOW TO DO THIS ?

In the case of my Direct Numerical Simulation, I have to resolve u_i + L_{jj} u_i = RHS(known) implicitly by matrix factorization to gain speed, but now it's a little bit more comlex. Thank for your help. O.DOCHE

It looks like a convection-diffusion equation. It is linear since v is known. State the problem in a weak formulation and use finite elements. For a quick solution try to use FreeFEM <http://www.freefem.org>

Yes it is, but I have to intgrate it to another fortran program so I can't use FEM. I've heard about ADI method (alternative direction impict), is it a good method for my problem ( quick resolution )?

Another method, the Full MultiGrid method ( found in The Numerica recipes ) but I have to implement it in a 3D case. Is it easily feasible ?