The two main elements of a multigrid solver are a relaxation scheme and a coarse grid correction. The relaxation is usually a local process that for elliptic problems it has the smoothing property. That is, high frequency errors decay fast as a result of relaxation. Low frequency errors, on the other hand, are slow to converge and an acceleration is required for them. The coarse grid correction serve exactly for that purpose. In standard multigrid method the coarse grid approximate the smooth part of the fine grid error. Since relaxation smoothes the error, applying a coarse grid correction after a few relaxation sweeps results in an effective reduction of the error for all components.
For nonlinear problems there is a version of multigrid called FAS (Full Approximation Scheme) that does not use linearization in the coarse grid correction. This version of multigrid will be used later for our optimization techniques.