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Numerical Approximation of PDE System Fixed Point Mappings via
Newton's Method

#### By: Joseph W. Jerome

Since the fundamental paper of Moser (Ann. Scuola Norm. Pisa
XX(1966), 265-315), 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 (Numer. Math. 47(1985),
123-138). It was proposed by the author (Numer. Math. 55(1989), 619-632)
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
(Approximate Solution of Operator Equations, Wolters-Noordhoff, 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.

This paper appeared in:
Journal of Computational and Applied Mathematics 38 (1991) 211--230.

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