We present a communication avoiding ILU0 preconditioner for solving large systems of linear equations (Ax=b) by using iterative Krylov subspace methods. Our preconditioner allows to perform s iterations of the iterative method without communication, by applying a heuristic alternating min-max layers reordering to the input matrix A, and through ghosting some of the input data and performing redundant computation. We also introduce a new approach for reducing communication in the Krylov subspace methods, that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration, based on the domain decomposition of the graph of A. The enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and to parallelizable algorithms with less communication, with respect to Krylov methods. We present several new Conjugate Gradient methods, such as the multiple search direction with orthogonalization CG (MSDO-CG), long recurrence enlarged CG (LRE-CG), and the short recurrence enlarged CG versions (SRE-CG, SRE-CG2, truncated SRE-CG2).