The objective of this work was to develop improved variations of the BEPS, FAC, and Sequential preconditioning methods used to solve composite grid linear systems. These methods are attractive since they use standard subgrid approximate factorizations in constructing the composite grid preconditioning. This paper describes two algebraic, multilevel grid coarsening procedures which allow efficient "black box" implementation of FAC and BEPS. The coarsening procedures are compared with regard to their effect upon convergence and computer efficiency. In addition all three methods may be modified to use variable step subgrid solutions to enhance convergence.