An accurate computation of the near-field enhancement is a key factor for the optimization of nanostructures in plasmonics. This problem has been addressed for Green's dyadic method but remains open for finite element method (FEM) where the use of non-Cartesian meshes is known to be the most efficient. We present a new adaptive mesh process based on the a posteriori error indicator estimation on the physical solution. This new procedure accelerates drastically the convergence of the solution and minimizes both the memory requirement and the computational time.