  In this work, we propose a multigrid preconditioner for Jacobian-free
Newton-Krylov (JFNK) methods. Our multigrid method does not require to store
any representation of the Jacobian at any level of the multigrid hierarchy. As
it is common in standard multigrid methods, the proposed method also relies on
three building blocks: transfer operators, smoothers, and a coarse level
solver. In addition to the restriction and prolongation operator, we also use a
projection operator to transfer the current Newton iterate to a coarser level.
The three-level Chebyshev semi-iterative method is employed as a smoother, as
it has good smoothing properties and does not require the representation of the
Jacobian matrix. We replace the direct solver on the coarsest-level with a
matrix-free Krylov subspace method, thus giving rise to a truly Jacobian-free
multigrid preconditioner. We will discuss all building blocks of our multigrid
preconditioner in detail and demonstrate the robustness and the efficiency of
the proposed method using several numerical examples.
