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Jacobian-free Newton-Krylov methods: a survey of approaches and applications. (English) Zbl 1036.65045

Summary: Jacobian-free Newton-Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means.
Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the JFNK method to any given problem is dependent on adequate preconditioning. JFNK has potential for application throughout problems governed by nonlinear partial differential equations and integro-differential equations.
In this survey paper, we place JFNK in context with other nonlinear solution algorithms for both boundary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of JFNK and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that JFNK can be wrapped (as an accelerator) around another nonlinear fixed point method (interpreted as a preconditioning process, potentially with significant code reuse).
The aim of this paper is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how JFNK may be applicable to applications of interest and to provide sources of further practical information.

MSC:

65H10 Numerical computation of solutions to systems of equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65R20 Numerical methods for integral equations
35Q35 PDEs in connection with fluid mechanics
45G10 Other nonlinear integral equations
65Z05 Applications to the sciences
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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