Question about multigrid method theory
 

Hi, everyone

There's one thing i don't understand about the multigrid method. I'm wondering why we smooth our residual from fine to coarse? It's part of updating the equation. Why don't we calculate the residual just at the coarse mesh?

Thanks!

Best,
lnk

the change of grid and the computational stencil can be seen as a sort of "filtering process" for the Fourier components of the residual...the high wavenumbers are rapidly smoothed in this way

Does it mean if I calculate the residuals of fine mesh then use the obtained fine mesh residuals to calculate the residuals for coarse mesh, the residual for coarse mesh can be more accurate than calculated directly from the coarse mesh?

I'm not sure I understand exactly what you mean ... going from fine (hf) to coarse (hc) grid you have automatically a grid filter (cut-off) at the corresponding Nyquist frequency pi/hc< pi/hf . Therefore less components (at high wavenumbers) of the errors. Additional smoothing of the Fourier components can be in effect depending on the algorithm ...
Maybe someone else can better address the issue ...

From what I understand the typical smoother will quickly reduce high frequency errors on a grid but will slowly reduce low frequency errors. The idea then is to remove the high frequency errors on a fine grid with a few sweeps of a smoother then make the grid course so that the low frequencies on the fine grid now appear to be high frequencies which can be quickly reduced. The process repeats

The idea is to represent the fine level problem on as coarse as possible coarse level. Physically if the curve is smooth you can represent it using little number of points but if it has local features you can not represent it with low number of points. Similar principle applies to multigrid too. You want to create as small coarse problem as possible to save cost.

Further a good smoother is one that would smooth it with as little iteration as possible.

Also note that what looks physically smooth to you may not be algebraically smooth to multigrid.

I would strongly suggest to buy this book if really serious about learning multigrid.

https://computation.llnl.gov/casc/pe...t/welcome.html


There are free notes also available but book is worth much much more than what you might pay for it. I bought this book recently even after reading all the free notes of this available and it taught me a lot.

I have question about applying Multigrid method in SIMPLER algorithm in collocated grid. In fact, for calculating mass flux in coarse grid we should corrected with mass flux correction and mass flux correction is calculated by using Rhie and Chow interpolation for the difference between the current approximation Uk and the restricted approximation Uk.
this is the way that some papers and books, for example Peric, are introduced. but I can not understand what exactly should be done. in fact, Rhie and Chow method we calculated face velocity with two terms, a pressure term and pseudo velocity. in second term we have source term that contains velocity. my question is if this source term should calculated by difference velocity or current approximation Uk. and also for pressure term, if it shoud be calculated with current approximation pressure or difference between the current approximation pressure and the restricted approximation pressure?
I hope I can convey my purpose.

Thanks in advance

I think you are little bit confused. To make things clearer :

1. You are talking about Full Or Geometric multigrid. In principle they are same as algebraic multigrid (some way) but they are really different.

2. In full multigrid, you have to descretize equations again at coarser levels.

3. For momentum and pressure equations you will need mass fluxes. This is where you main confusion is.

4. Mass fluxes at the coarser levels could be computed by just summing the fluxes enclosing coarser control volume by using fine level fluxes. (think about it).

5. Once momentum equations are solved you get fine level fluxes and again coarser level fluxes are calculated as mentioned in (4).

6. The real challenge of full multigrid is to desretize on coarser level.

7. Multigrid for u,v,w is rarely needed (because of diagonal dominance).

Dear Arujan;
Thank you for your answer. in fact, I Used FAS multigrid in NS equation, and I know equation should be descritized in coarser grid. in fact in coarse grid a source term is added to solve equations in fine grid instead of coarse grid, and also in calculating mass flux in coarse grid, restricted mass flux claculated by sum of the faces of the child grids that make the parent's face. and then rectify in each iteration in coarse grid by difference between the current approximation velocity and the restricted approximation velocity.
as I mention in my question, my problem is calculating mass flux correction in Rhie and Chow method by this difference velocity. in rhie and chow method, how should be behaved for velocity and pressure. does pressure should be approximation pressure or difference between the current approximation pressure and the restricted approximation pressure? and also in in source term of pseudo velocity term. my problem is calculating face velocity in coarse iteretion by the velocity that calculated difference between the current approximation velocity and the restricted approximation velocity.

I am afraid I can not be much help there. Because I never applied Full multigrid to Navier Stokes and thus never had to solve for velocity at coarser levels.

I did however apply full multigrid to pressure poisson problems on cartesian type grid system.

I do have a book at home that might have answer to your question, i will look in it and let you know.

Dear arujan;
Thank you so much. I'm looking forward to hearing from you.

Regards;
Mehdi

sorry I could not find anything useful for you.
i have this book ,
http://books.google.de/books/about/m...IC&redir_esc=y

which has collection of papers where people used multigrid but i could not find anything that explains what you are looking for.

However please have a look at
http://www.tfd.chalmers.se/~lada/

there read

JOHANSSON, P.
"A Three-Dimensional Laminar Multigrid Method Applied to the SIMPLEC Algorithm", MSc thesis, Rept. 92/5, Thermo and Fluid Dynamics, Chalmers University of Technology, Göteborg, 1992.


This is not quite what you asked but it is interesting read.

Dear Arujan;
Thank you so much for your help. I already read the last one, it is really helpful and solved some of my problems. but unfortunately, I can not get the formers. I try to obtain it, and I hope it can fix other problems.

best regards;
Mehdi