The standard PTMG method, invented and tested for the inversion of the fermion matrix (Dirac-operator) in the case of U(1) and SU(2) lattice gauge theories with staggered fermions, has now been generalized to SU(3). A crucial ingredient of the new algorithm is the way SU(3) link elements are averaged in the generation of the coarse grid link elements. Numerical results are presented for the two-dimensional case. For 128 × 128 lattices and physically relevant values of the coupling constant β and of the quark mass mq, the new algorithm is an order of magnitude faster in CPU time than its leading competitor, the Conjugate-Gradient algorithm.