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.
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