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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