Multigrid : Restriction operator

[kolmog]

Hello everybody,

I have a small question related to multigrid. I am trying to solve a Poisson equation in 3D with periodic boundary conditions with cell-centered multigrid. I have programmed a quite fast serial code which performs V-cycles with a maximum of 7 grids. For the interpolation from a coarse grid to a fine grid, I have used trilinear interpolation (that in page 69 of the book "An introduction to multigrid methods (Wesseling)"). My doubt deals with the choice of the restriction operator. In the case of using bilinear interpolation (2D) or trilinear interpolation in 3D, the most common choice is "full weighting" for the restriction operator. In my case I have adopted the following formula:

F_i=$\frac{1}{8}$Ʃ$^{8}_{j=1}$f_j

with 'f' being the value of a function in the fine grid (spacing = h) and F being the value in the coarse grid (H=2h). That is, in my cell-centered multigrid code, values in the coarse grid, are generated by taking into account the first 8 fine grid neighbors surrounding the point in the coarse grid.

I think it is a reasonable choice, but taking into account that I use Galerkin approximation to compute the left-hand side operator in the different coarse grids, I want to be sure. I am considering a fourth-order (Mehrstellen) scheme for the discretization of the left-hand side oprator in the finest grid).

I would appreciate a response.

 

[jvicens]

[Username]

Dear [Username],

Thank you for your question. It is great to see that you are working with multigrid methods to solve a Poisson equation in 3D with periodic boundary conditions. Your approach of using trilinear interpolation for the interpolation and a combination of bilinear/trilinear interpolation for the restriction operator seems reasonable. However, I would suggest considering using a full weighting restriction operator instead of your current choice.

The full weighting restriction operator takes into account all neighboring fine grid points, rather than just the first 8 neighbors. This can lead to a more accurate approximation and better convergence rates. Additionally, using a fourth-order scheme for the discretization of the left-hand side operator in the finest grid is a good choice, as it can also improve the accuracy of your solution.

I would also recommend testing your code with different restriction operators and comparing the results to see which one gives the best convergence rates and accuracy for your specific problem. Additionally, you may want to consider using a multigrid method with adaptive mesh refinement, as it can further improve the accuracy and efficiency of your solution.

I hope this helps and good luck with your research!