An Asymptotically Linear Fixed Point Extension of the Inf-Sup
Theory of Galerkin Approximation
By: Joseph W. Jerome
Babuska and Aziz
introduced a Galerkin approximation theory for saddle point
formulations of linear partial differential equations (The Mathematical
Foundations of the Finite Element Method
with Applications to Partial Differential Equations, Academic Press, 1972).
a powerful extension of the approximation theory for positive-definite,
Independently, a coherent theory for the approximation of fixed points
of nonlinear mappings
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 shown to be a logical
extension of the inf-sup theory constructed by
Babuska and Aziz. In the process, we obtain sharp lower bounds,
not emphasized by these authors. We also identify a novel fundamental
approximation property of the nonlinear calculus, which we characterize
as asymptotic linearity. This is adjoined to a robust estimate for
replacement of numerical fixed points in the outer iteration
by a single Newton inner iteration,
beginning each time with the previously computed
outer iteration numerical fixed point.
This yields a tight closure to the property of asymptotic linearity.
This paper appeared in Numer. Funct. Anal. Optim. 16 (1995), 345--361.
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