## An Asymptotically Linear Fixed Point Extension of the Inf-Sup
Theory of Galerkin Approximation

#### By: Joseph W. Jerome

In 1972,
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).
It represented
a powerful extension of the approximation theory for positive-definite,
self-adjoint operators.
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.

It can be viewed in the following format: