How to combine multigrid preconditioner with jacobi preconditioner?

Question

I have not found any relevant information in the literature on the following rather simple problem:

How to combine (geometric) multigrid preconditioned conjugate gradient (MGPCG) with an additional jacobi preconditioner ?

Background: Large (symmetric, diagonally dominant) linear system A*x=b arising from discretized incompressible two-phase flow resulting in a variable coefficient poisson equation where the very high contrast (1:1000) of coefficients leads to high condition number of matrix A and thus poor convergence of the multigrid preconditioned CG. In my case convergence is very good (factor 10 residual reduction per MGPCG iteration) as long as the matrix coefficient contrast is low (<1:10) but deteriorates quickly with increasing contrast. From what I read so far, the condition number of such an ill-conditioned matrix can be improved via diagonal scaling, i.e. Jacobi preconditioner. So the question is how to combine the multigrid preconditioner with an "additional jacobi preconditioning"

Would it be sufficient to simply apply the usual diagonal scaling using D_l = diag(A_l) at each level l of the MG fine-to-coarse hierarchy of Matrices A_l, i.e. at each level solving D_l^(-1/2) * A_l * D_l^(-1/2) * y = D_l^(-1/2) * b_l instead of A_l*x=b_l (Using the same prolongation & restriction operators as for the original Problem ?)

Answer 1

The way to integrate "additional" preconditioners into multigrid is via the smoother. At every level of the multigrid algorithm, you have to pre- and post-smooth the residual, and that is often done using stationary solvers such as Jacobi, Gauss-Seidel, or similar. If you have a preconditioner that works well for problems with variable coefficients, then it will also serve as a good smoother in the multigrid method.