AN ASYMPTOTICALLY LINEAR FIXED POINT EXTENSION OF THE INF-SUP THEORY OF GALERKIN APPROXIMATION

In 1972, Babuska and Aziz introduced a Galerkin approximation theory for saddle point formulations of linear partial dierential equa-tions (The Mathematical Foundations of the Finite Element Method with Applications to Partial Dierential Equations, Academic Press, 1972). It represented a powerful extension of the approximation theory for positive-denite, self-adjoint operators. Independently, a coherent theory for the approximation of xed points of nonlinear mappings by numerical xed points was devised by Krasnosel'skii and his cowork-ers (Approximate Solution of Operator Equations, Wolters-Noordho, 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 xed points in the outer iteration by a single Newton inner iteration, begin-ning each time with the previously computed outer iteration numerical xed point. This yields a tight closure to the property of asymptotic linearity. Dedicated to the memory of Farouk Ode