I have a question related to the Multigrid method applied to say Euler solutions. One of the oft quoted advantages is that the high frequency errors are quickly damped by this process, leading to accelerated convergence rates.

My question is whether it is necessary to have multiple grids inorder to be able to do this. Has anyone developed a method by which the errors are split by discrete Fourier Analysis into varying frequency bands, and each error band is successively removed as the solution progresses ? This way, it should be possible to accelerate convergence on any grid, using just one grid. One may then refine the grid in regions of high gradients and compute another solution on this new grid, and so on.

(1). I am not on the multi-grid method side, because the programming is more complicated. I love the simple method, and I think the computer should take care of the rest of it(hard work). (2). If one can obtain solution directly without iteration, then you need only one grid. (3). If you have to obtain the solution iteratively through grid point, then each time the information can propagate is the size of the computational template. So, if the template size is physically small (many grid points, fine mesh), the local exchange of information will quickly smooth out the errors, but not the long waves because the information is still sitting at far away places. (4). There are probably many ways to program the multi-grid to speed up the information propagation, but still, I think the best way to do is to try the direct method, or something like that. (or semi-direct method).

I don't have a precise answer to your question, but let me say something.

Low-frequency-error is the problem because it takes so long to damp them out (i.e. there exist iterative methods that is very good at eliminating high-frequency error modes). The main point of multi-grid method is to damp this low-frequency error on a grid coarser than the original grid, where the low-frequency error becomes high-frequency which can be damped out quickly. So, project the error (or sol) on a coarser grid seems essential to the multi-grid method.

Now, can we do this on a single grid? One way to kill low-frequency mode without using any other grids would be to devise a smoother (iterative solver) which is very effective especially for low frequency modes. And then, maybe, we can improve the convergence by using two different iterations alternatively (kill high-freq. and kill low-freq.).

I guess, the main point is whether or not we can devise an interative method that is effective equally for all the frequency modes. Oh, it sounds like preconditioning, doesn't it?

Hi,

You're not wrong - multigrid can be done without a hierarchy of grids - in fact, you don't even need a grid because multigrid can be applied directly on the system of algebraic equations. Also, you don't need anything as fancy as Fourier analysis - the solver is actually pretty simple. The idea is called Algebraic Multigrid (AMG) and if you look through the web you'll find a lot of references.

Regards,

Hrv