A Multigrid Preconditioned Newton--Krylov Method

Abstract

We study multigrid preconditioning of matrix-free Newton--Krylov methods as a means of developing more efficient nonlinear iterative methods for large scale simulation. Newton--Krylov methods have proven dependable in solving nonlinear systems while not requiring the explicit formation or storage of the complete Jacobian. However, the standard algorithmic scaling of Krylov methods is nonoptimal, with increasing linear system dimension. This motivates our use of multigrid-based preconditioning. It is demonstrated that a simple multigrid-based preconditioner can effectively limit the growth of Krylov iterations as the dimension of the linear system is increased. Different performance aspects of the proposed algorithm are investigated on three nonlinear, nonsymmetric, boundary value problems. Our goal is to develop a hybrid methodology which has Newton--Krylov nonlinear convergence properties and multigrid-like linear convergence scaling for large scale simulation.