Hi, every body I've implemented a 3D Multigrid with Neumann and Periodic Boundary condition. I solve a Poisson equation on an irregular mesh of 65*65*65 with finite difference scheme.

I obtain convergence in 12 MultiGrid cycles. Is it a good performance or not ??

PS: I do not use yet the 'Nested iteration'algorithm

That could depend on what is your tolerance of the convergence.

This is obviously true. I have impose eps=1.E-7. with a V-cycle.

it sounds good.

Hi Davoche! I am also solving poisson equation with almost similar number of nodes. But I am solving with SOR method and nearly 500 iterations are required for convergence.(1.0E-5) Since your solution is getting converged in 12 cycles only, there may be something wrong. Please check the velocity componants used as input in the poisson equation. What about the results?? regards, Jas

Jas,

That's the main reason to use multigrid - it accelerates convergence. The improvement over SOR is usually dramatic, so maybe you should consider changing your solver.

Rami

For a uniform grid and a smooth solution the answer would be no. A typical number here would be about half that for a properly sorted scheme. Even for a FAS scheme (solving u, v, w & p) for a low Reynolds number driven cavity 6 or 7 iterations would be typical for a sorted scheme.

The important property is less the number of iterations but that it requires the same number on 16^3, 32^3, 64^3, 128^3,...

However, when the solution is not smooth and/or your coefficients vary strongly in size (i.e. non-uniform grid) then the performance will drop. So your 12 iterations may be fine but you need to check by running on different grid sizes. Often you can bring back lower numbers of cycles by adopting a better smoother and/or smoothing strategy. Unfortunately, the cost in CPU of such changes is not always worthwhile.

Just a general question to those who are familiar with multigrid. Can you apply a multigrid or multilevel solver to a general linear system of equation, those otherwise could be solved with Gauss-Seidel etc. I mean lets say its not related to cfd, but just a set of linear equation of the form Ax=Q, can we apply this kind of strategy to accelerate this convergence.

It depends on the nature of the equations being solved. Multigrid requires a smoother (a fast simple approximate solver) that strongly reduces local errors so that when you run through the heirarchy of grids all error scales get strongly reduced. The solution also needs to be smooth for a general scheme although there are things one can do to adapt for specific cases.

Wave equations for example do not work out of the box with multigrid. However, transforming the equations from the time domain can produce something that multigrid handles well.

Multigrid offers the ideal of scaling linearly with problem size and is often the only viable approach for solving implicit equations with many millions or billions of unknowns. Because of this, most sets of equations have people somewhere working to get multigrid going if it does not work when applied directly.

Hi everybody, -> Parth: Sorry but I don't undersatnd your question, can you reformulate it, please.

-> Andy: I've tried with different grid size and I have always 12 cycles. But something is very strange. I have even 12 cycles with a UNIFORM grid. Maybe this is due to my Restrication and Prolongation operators. In fact, the transfer from one grid to another is made regularly ( no anisotropic coef ). I have try to implement it but the CPU cost is increased and I have always 12 cycle. No idea about that so I'm trying to use Nested iteration now (Full Multigrid ).

My question was that for a given set of linear equations say AX=q ; and it should not matter how i got them or not neccessarily from CFD formulations, they could just be a set of linear equation. To this set of linear equations can we apply the acceleration due to multigrid kind of scheme.

the reason for asking so was this, i have seen mostly that we apply multigrid for the equations resulting from decretisation of partial differential equations. My question was can we apply it to system of such equations NOT resulting from partial differential equations. Does any such scheme exists.

(i am no expert with multigrids so i think people who know better would be able to shed some light on this).