  In this work, we propose a multigrid preconditioner for Jacobian-free
Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge
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.
