  We introduce a new numerical method to approximate the solutions of a class
of stationary Hamilton-Jacobi (HJ) partial differential equations arising from
minimum time optimal control problems. We rely on nested grid approximations,
and look for the optimal trajectories by using the coarse grid approximations
to reduce the search space in fine grids. This provides an infinitesimal
version of the ``highway hierarchy'' method which has been developed to solve
shortest path problems (with discrete time and discrete state). We obtain, for
each level, an approximate value function on a sub-domain of the state space.
We show that the sequence obtained in this way does converge to the viscosity
solution of the HJ equation. Moreover, for our multi-level algorithm, if
$0<\gamma \leq 1$ is the convergence rate of the classical numerical scheme,
then the number of arithmetic operations needed to obtain an error in
$O(\varepsilon)$ is in $\widetilde{O}(\varepsilon^{-\theta })$, with $\theta<
\frac{d}{\gamma}$, to be compared with $\widetilde{O}(\varepsilon^{-d/
\gamma})$ for ordinary grid-based methods. Here $d$ is the dimension of the
problem, $\theta $ depends on $d,\gamma$ and on the ``stiffness" of the value
function around optimal trajectories, and the notation $\widetilde{O}$ ignores
logarithmic factors. In particular, in typical smooth cases, one has $\gamma=1$
and $\theta=(d+1)/2$.
