On Using Approximate Finite Differences in Matrix-Free Newton-Krylov Methods

A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton’s method. Newton–Krylov methods are often implemented in “matrix-free” form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approximate function values in these finite differences. We first formulate a finite-difference Arnoldi process that uses approximate function values. We then outline a Newton– Krylov method that uses an implementation of the GMRES or Arnoldi method based on this process, and we develop a local convergence analysis for it, giving sufficient conditions on the approximate function values for desirable local convergence properties to hold. We conclude with numerical experiments involving particular function-value approximations suitable for nonlinear diffusion problems. For this case, conditions are given for meeting the convergence assumptions for both lagging and linearizing the nonlinearity in the function evaluation.