Multigrid (MG) may be used to solve a linear system $Ax=b$ by constructing an initial guess $x_0$ and repeating the following for $i=0,1..$ until convergence:
- Compute the residual $r_i = b-Ax_i$
- Apply a multigrid cycle to obtain an approximation $\Delta x_i \approx e_i$, where $Ae_i = r_i$.
- Update $x_{i+1} \gets x_i + \Delta x_i$
The multigrid cycle is some sequence of smoothing, interpolation, restriction, and exact coarse grid solve operations applied to $r_i$ to produce $\Delta x_i$. This is typically a V-cycle or a W-cycle. This is a linear operation so we write $\Delta x_i = B r_i$.
One can interpret this process as preconditioned Richardson iteration. That is, we update $x_{i+1} \gets x_i + B r_i$.
Richardson iteration is a prototypical Krylov subspace method, which suggests the use of multigrid cycles to precondition other Krylov subspace methods. This is sometimes called "accelerating" multigrid with a Krylov method, or alternately can be seen as a choice of a preconditioner for a Krylov method.
Another way to extend the algorithm above is to employ Full Multigrid (FMG). See this answer for a concise description.
In what situations is Krylov-accelerated MG preferable to MG or FMG?