Will a multigrid method damp out the high frequency physical wave?
I am solving timedependent problem, which contains physical waves with different frequencies. The NS equations I use contain a Poisson's equation. I use iterative method to solve it, but I find the convergence speed is very low.
Hence, I hope to use a multigrid method to accelerate the convergence. However, I am afraid that the high frequency physical waves cannot be represented in a coarse grid and my simulation target will be damped out. Will this happen? By the way, I feel rather confused about the concept of "low frequency errors" in multigrid method. Is it the same thing as the physical waves with low frequency I need to simulate? Thank you in advance. 
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No. The coarser grids are just used to get an error estimate that is then "prolonged" back to the fine grids. In the end you still have a solution on your fine grid that includes the higher resolution information. Quote:
Not really; lots of iterative methods (eg. point GaussSeidel, SSOR) are good at getting rid of high frequency error, but take many iterations to smooth out the lowfrequencies. By going to a coarse grid, what used to be low frequency errors on the fine grid become higher frequency (which the method handles well), so convergence speeds up. Quote:
HtH, Josh 
Hi,
I have encountered the same problems as you. Do you mean that your MG doesn't work well? I have experiences that at first I checked up MG's procedure by programming 1D problem and then challenged 3D one which was my aim. http://www.geocities.jp/viola08129/ 
Thank you all very much.
I have programmed the multigrid method. However, I really find the wave amplitude is 10% smaller by multigrid method than only using a fine grid. Is this a normal result of multigrid method? By the way, when solving the Poisson’s equation, I didn’t use the restriction operatior on the source term. Is it necessary? Thank you. 
restriction operatior etc.
Hi,hadesmajesty
[QUOTE=hadesmajesty;230777]Thank you all very much. I have programmed the multigrid method. However, I really find the wave amplitude is 90% smaller by multigrid method than only using a fine grid. Is this a normal result of multigrid method? It isn't a normal result, I think.:confused: By the way, when solving the Poisson’s equation, I didn’t use the restriction operatior on the source term. Is it necessary? Necessary for rsiduals(source term?). V cycle procedure I used with max. grid level 3 for linear equation system(1D problem) Aw=G is as follows; w=0 ! initial solutions f[3]=G ! initial residuals at max. grid level=3 do i=1,niter ! n in [n]:grid level f[2]=R[2,3]*f[3] f[1]=R[1,2]*f[2] u[3]=0 ! initial residual at g.l.=3 u[3]=relax(A[3]*u[3]=f[3]) ! relaxation for residual eq. at g.l.=3 f[2]=R[2,3]*(f[3]A[3]*u[3]) ! restriction for residuals f u[2]=0 u[2]=relax(A[2]*u[2]=f[2]) f[1]=R[1,2]*(f[2]A[2]*u[2]) u[1]=inv(A[1])*f[1] ! exact solution at g.l.=1 u[2]=u[2]+I[2,1]*u[1] ! prolongation and correction u[3]=u[3]+I[3,2]*u[2] f[3]=f[3]A[3]*u[3] w=w+u[3] ! correction for solution end do Thank you. 
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Thank you very much, jstults and trout. I think I have found what’s wrong.
To solve Ax=b. Last time I used the iteration scheme like, x(n)=G*x(n1)+b, where G and b are fixed in all iterations, and just transferred the unknown x and the source term b between the coarse and fine grids. Now I used the iteration scheme: Residual=(A*x(n1)b) dx=G’* Residual x(n)= x(n1)+dx I transfer the residual (A*x(n1)b) between the two grids. The multigrid cycle is that: first sweep two times on the fine grid, then transfer and sweep one time on the coarse grid, and then go back to fine grid. This time I find the result is the same as just using fine grid. However, the above algorithm has larger computational cost, because I have to do matrix multiplication for two times : first A*x(n1) for the residual and second G’* Residual for dx. Therefore, the final computational times are as below: 1. directly solve x only on fine grid: 4.46s.(my old method) 2. Solve residual and then dx only on fine grid: 6.73s. 3. Solve residual and then dx by multigrid: 4.61. So, compare the time used by 1. and 3. The multigrid method cannot improve anything. Is my current algorithm correct? Or is there any place to improve? Thanks again. 
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The early slides are also good, they talk to your original questions about the low/high frequency errors. 
Thanks a lot.
I follow the book of William Briggs and use a twolevel coarse grid correction method. The solution is correct, but the convergence speed has not been improved or even lower. The Poisson's equation I solve is a*d^2u/dx^2+d^2u/dy^2=f, in which 0<a<<1. Hence, I use GaussSeidel line iteration by solving variables in y direction at the same time. In the coarse grid correction step, according the book of William Briggs, I just coarsen the grid in y direction. For each line, I calculate the residual, then the error and update the new value. After that I go to next line and repeat the above procedure again. I am not sure how to implement the coarse grid correction method with the GaussSeidel line iteration. Is my algorithm correct? Thank you very much. 
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