Usually , in a simulation of an uncompressible fluid, it used a CFD technique (finite differences, volume,etc..), anyway the physical domain of the fluid is divided in cells. It is also necessary to solve the Poisson equation for the pressure, wich rapresent the most slow calcuus process. I wish to know wich is the dependence of the computational complexity from the number of cells in the domain (N^2,NlogN?) Thanks

(1). How do you define the computational complexity? (2). Since it is necessary to study the mesh-independence issue of the solution, the number of cells must be increased in the study. With more cells to solve, it will take longer to compute the flow field. (3). The convergence of the solution depends on the methods used, such as the point-iterative method, the line-iterative method or the direct solution method. It probably depends on the problem itself, I think.

1) I define it so: the number of step wich the machine need to do ( after one time step ) to calculate the pressure in all domain (for every cell). If we want to calculate the pressure in one cell , in a Poisson equation, we need to know the values of pressure on all boundary, yes? For example in a box , I think in the worst case, if we divide the domain in N cells , we shoul need N^2 step for every time step. Maybe this is improved by use of particular techniques. Anyway, isn't there an inferior limit ? I know it depends on the problem, but in a general case of a box where I solve a Poisson equation?

(1)For every time step, you have to update the pressure at every cell. Since you have divided the domain into N cells, you have to update the pressure at N cells, right? Why N^2 steps?

You do not need the pressure on the boundary, if you know teh velocities. it depends on the the method you are using, mostly you could use dp/dn = 0, when teh velocities are known.

If you'll solve your problem in (u,v,w,p) then you should solve Neimann problem for Poissonn equation on every time step i. But if you'll solve in stream functions and etc lang then you should solve Neimann plus Poisonn only one. Of course boundary conditions will be more comlex.

Also you can solve Neimann problem with correction on every time step i: $P^i- \int_{Omega} P^i$ then you will have only CNlogN operations.