SIAM Journal on Numerical Analysis
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.
Brown, P. N., Walker, H. F., Wasyk, R., & Woodward, C. S. (2008). On Using Approximate Finite Differences in Matrix-Free Newton-Krylov Methods. SIAM Journal on Numerical Analysis, 46(4), 1892-1911. http://dx.doi.org/10.1137/060652749
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