A Multigrid Preconditioner for Jacobian-free Newton-Krylov Methods

TL;DR

This paper introduces a Jacobian-free multigrid preconditioner for Newton-Krylov methods that does not require Jacobian knowledge, enhancing efficiency and robustness in solving nonlinear systems.

Contribution

It presents a novel multigrid preconditioner that is fully Jacobian-free, utilizing a projection operator and Chebyshev smoothing, with demonstrated numerical efficiency.

Findings

Demonstrates robustness across multiple numerical examples

Achieves efficient convergence without Jacobian matrix assembly

Provides detailed analysis of multigrid components

Abstract

In this work, we propose a multigrid preconditioner for Jacobian-free Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge of the Jacobian at any level of the multigrid hierarchy. As it is common in standard multigrid methods, the proposed method also relies on three building blocks: transfer operators, smoothers, and a coarse level solver. In addition to the restriction and prolongation operator, we also use a projection operator to transfer the current Newton iterate to a coarser level. The three-level Chebyshev semi-iterative method is employed as a smoother, as it has good smoothing properties and does not require the representation of the Jacobian matrix. We replace the direct solver on the coarsest level with a matrix-free Krylov subspace method, thus giving rise to a truly Jacobian-free multigrid preconditioner. We will discuss all building blocks of our multigrid preconditioner in detail and demonstrate the robustness and the efficiency of the proposed method using several numerical examples.