This GPU based script draws u cross-section after multigrid V-cycle
with the reduction _{i,n/4}level = 6 and "deep" relaxation iterations 2rel.
Boundaries are periodic
f
_{i,j} = sin(2πi/n) sin(2πj/n)n rel level ω |

Let

If we write weighted Jacobi operator as

then error reduction equation is

Weighted Jacobi relaxation smooths the error but low frequencies are undamped. On the coarse grid, however, this error appears more oscillatory, and relaxation will be more effective. Therefore we go to a coarse grid and relax on the residual equation

**Multigrid V-cycle**

- two smoothing Jacobi iterations on
*i*grid^{th} - compute
*i*residual^{th} - reduce residual on coarse
*i+1*grid^{th} *2rel*"deep" Jacobi iterations on the coarsest grid*i*error is interpolated on fine^{th}*i-1*grid and is used for^{th}*i-1*error correction^{th}- two smoothing Jacobi iterations on fine
*i-1*grid^{th}

[1] William L. Briggs, Van Emden Henson, Steve F. McCormick A Multigrid Tutorial (look at "Tutorial Slides")

Simulations on GPU