  We develop the max-plus finite element method to solve finite horizon
deterministic optimal control problems. This method, that we introduced in a
previous work, relies on a max-plus variational formulation, and exploits the
properties of projectors on max-plus semimodules. We prove here a convergence
result, in arbitrary dimension, showing that for a subclass of problems, the
error estimate is of order $\delta+\Delta x(\delta)^{-1}$, where $\delta$ and
$\Delta x$ are the time and space steps respectively. We also show how the
max-plus analogues of the mass and stiffness matrices can be computed by convex
optimization, even when the global problem is non convex. We illustrate the
method by numerical examples in dimension 2.
