Parallel Matrix-free polynomial preconditioners with application to flow simulations in discrete fracture networks

We develop a robust matrix-free, communication avoiding parallel, high-degree polynomial preconditioner for the Conjugate Gradient method for large and sparse symmetric positive definite linear systems. We discuss the selection of a scaling parameter aimed at avoiding unwanted clustering of eigenvalues of the preconditioned matrices at the extrema of the spectrum. We use this preconditioned framework to solve a $3 \times 3$ block system arising in the simulation of fluid flow in large-size discrete fractured networks. We apply our polynomial preconditioner to a suitable Schur complement related with this system, which can not be explicitly computed because of its size and density. Numerical results confirm the excellent properties of the proposed preconditioner up to very high polynomial degrees. The parallel implementation achieves satisfactory scalability by taking advantage from the reduced number of scalar products and hence of global communications.