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Joshua Lee March 14, 2000 01:30

Algebraic Multigrid for nonuniform grid

I have been trying to use Algebraic MG (developed by Brandt) in solving a flow around an obstacle (2D and 3D). I have found that AMG has superior convergence rate when the grid is relatively uniform, but shows poor convergence when nonuniform grid distribution is used. Theoretically, MG must have higher convergence rate than preconditioned conjugate gradient; however, it does not seem that way. The poorness of AMG is even bigger when the computational boundary is curvilinear.

Does anybody have any experience in this and want to give me some comments ?


Patrick Godon March 14, 2000 12:48

Re: Algebraic Multigrid for nonuniform grid
There are two ingredients in your problem that might be the source of the poor convergence.

First, if the grid is not uniform, then all the relaxation, restriction and prolongation processes have to be taken care of correctly, such that the center of the irregular grid cells is found (geometric center, etc..) to form the next grid level.

The other reason for the poor convergence might be due to the intrinsic property of the multigrid method itself. If you have a uniform grid at different levels: high resolution grid down to low resolution grid, then when you relax the solution on the coarse grid you actually smooth out the solution (or the error of it) on the long wavelengths (because of the large spacing between the points of the coarse grid). When you relax on a finer and a finer grid, you smooth out the errors on smaller and smaller wavelengths. All the 'trick' of the multigrid method is that you do not smooth out the long wavelengths on the fine grid, because this requires many many relaxations, but on each grid you smooth out the wavelenght that is characteristic to it. So on the fine grids the short ones and on the coarse grids the large ones.

Now if you do have a non-uniform grid, then at the same grid level you have actually different wavelengths and if you relax only a few times, you relax the longest wavelenghts only.

Any other hint?


Bob Anderson March 14, 2000 23:21

Re: Algebraic Multigrid for nonuniform grid
I'm not sure what you mean by "algebraic" multigrid - is this fundamentally different from the FAS Jameson type schemes? If not, your results are strange. In my experience multigrid really shines on stretched meshes where you'd have otherwise very slow propogation speeds. So, you are either talking about something I'm not familiar with, or you've made a mistake somewhere.


Joshua Lee March 16, 2000 05:03

Re: Algebraic Multigrid for nonuniform grid
Dear Bob:

I am using AMG that is fundamentally similar to the FAS Jameson type.

I got superior convergence rate and low CPU time compared to other solver for both uniform and nonuniform grids when AMG is used in Cartesian coordinate with the boundary aligned with coordinate. But when the boundary is curvilinear such as flow around a circular cylinder, AMG seems inefficient and the poor convergence becomes eminent when the number of grids increases (over to 0.3Millions). I wonder if you did get the same convergence rate as in the Cartesian grid when non-cartesian grid is used for curvilinear boundary surfaces.


Bob Anderson March 16, 2000 14:05

Re: Algebraic Multigrid for nonuniform grid

Our group has found that the multigrid scheme we use is very effective for curvilinear meshes. I don't know anyone who has made a comparison between performance on a cartesian mesh vs. a curvilinear mesh. What we normally look at is performance with multigrid vs. performance without multigrid on the same mesh, and in that case the speedup is usually dramatic. Can you try running your same code with the same mesh with only a single mesh level? That isolates your AMG method from everything else for purposes of comparison.


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