The ILU iteration scheme is well known as an excellent smoother in a multigrid process. But up to now a restricting fact of the method was that, apparently, the algorithm can only be applied efficiently to finite-difference discretizations on rectangular grids. The problem to transfer the algorithm to finite-element discretizations is that the iteration depends on the numbering of the grid points and on the structure of the grid. In opposition to this, the basic advantage of finite elements is that one can use self-adaptive refinement strategies, to get problem-orientated grids, which have not a uniform structure. In this paper we explain how to apply the ILU method to arbitrary finite-element grids and develop strategies for accelerating the algorithm and making it vectorizable. In Section 3 we shall study the influence of the grid for the stability of the ILU iteration and give a somewhat surprising example, which makes us optimistic with regard to a generalization of the theoretical results to larger classes of problems. Finally, in Section 4 we report on some numerical tests for an eigenvalue problem with real physical background.