#### Document Type

Article

#### Publication Date

4-18-2008

#### Publication Title

SIAM Journal on Numerical Analysis

#### Abstract

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.

#### Suggested Citation

Brown, Peter N.
, Walker, Homer F.
, Wasyk, Rebecca
, Woodward, Carol S.
(2008). On Using Approximate Finite Differences in Matrix-Free Newton-Krylov Methods. *SIAM Journal on Numerical Analysis, 46*(4), 1892-1911.

Retrieved from:
http://digitalcommons.wpi.edu/mathematicalsciences-pubs/58

#### Volume

46

#### Issue

4

#### First Page Number

1892

#### Last Page Number

1911

#### DOI

10.1137/060652749

#### Publisher Statement

© 2008, SIAM Publications.