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

TL;DR

This paper introduces a scalable, matrix-free polynomial preconditioner for the Conjugate Gradient method, improving the efficiency of large-scale flow simulations in discrete fracture networks.

Contribution

It presents a novel high-degree polynomial preconditioner that is communication-avoiding and matrix-free, tailored for large sparse systems in flow simulations.

Findings

Numerical results show excellent preconditioner performance at high polynomial degrees.

The parallel implementation scales well with reduced global communication.

Effective preconditioning accelerates convergence in flow simulations.

Abstract

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.