# How do multigrid approaches deal with Gibbs phenomenon?

I know (from https://scicomp.stackexchange.com/a/31339/20545, among others) that I need a certain mesh density in FEM, else I might get non-physical oscillations in my solution.
How do multigrid approaches deal with such problems? They reduce the problem to a coarser mesh, solve it there, and use the result (by smoothening) for either solving the problem directly on the fine mesh, or for preconditioning. While solving on the coarse mesh, those methods will result in oscillations. Will those be smoothened out during the process back to the fine mesh, or are there other restrictions which avoid bringing those oscillations back to the fine mesh?
Or is my understanding of the method wrong?

• I was considering a Galerkin method applied to a convection-diffusion problem. My doubt was about smoothing properties of the iterative scheme. Suppose $h$ is finest mesh on which mesh Peclet < 2 so $A_h$ is a good matrix (M-matrix?). Suppose on $2h$ mesh, mesh Peclet > 2 so that $A_{2h}$ is not an M-matrix. Would the iterations using $A_{2h}$ actually reduce the high frequency errors on $2h$-mesh ? @wolfgang-bangerth Aug 2, 2019 at 10:13