### Abstract

Since the fundamental paper of Moser (1966), it has been understood analytically that regularization is necessary as a postconditioning step in the application of approximate Newton methods, based upon the system differential map. A development of these ideas in terms of current numerical methods and complexity estimates was given by the author (1985). It was proposed by the author (1989) to use the fixed-point map as a basis for the linearization, and thereby avoid the numerical loss of derivatives' phenomenon identified by Moser. Independently, a coherent theory for the approximation of fixed points by numerical fixed points was devised by Krasnosel'skii and his coworkers (1972). In this paper, the Krasnosel'skii calculus is merged with Newton's method, for the computation of the approximate fixed points, in such a way that the approximation order is preserved with mesh independent constants. Since the application is to a system of partial differential equations, the issue of the implicit nature of the linearized approximation must be addressed as well.

Original language | English (US) |
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Pages (from-to) | 211-230 |

Number of pages | 20 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 38 |

Issue number | 1-3 |

DOIs | |

State | Published - Dec 23 1991 |

### Keywords

- Krasnosel'skii calculus
- Nonlinear systems
- approximate Newton methods
- finite-element methods
- fixed-point approximation

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics