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This assumption is quite natural because x* is the solution of the nonlinear equation (1.1) and Ax k is the solution of the linear equation (1.3), and therefore under reasonable conditions they have better smoothness properties. Let 7Tj : X —> Xj, j = 0,1,... be some family of linear projection operators. Assume that these operators are uniformly bounded on W*, which means that they satisfy a stability condition of the type \\n jX \\ < £||:r||, x € W\ qj < q < oo . (2.7) The projection property irj = TTJ then immediately implies that qj > 1 . (2.8) Subsequently, the most interesting case will turn out to be that the constants qj satisfy Jim qj = q. (2.9) J-fOO The fact that the subspaces Xj can be used to approximate the elements of W* well, at least asymptotically, is expressed by the approximation condition \\x-ir jX \\ < 8j\\x\\, IGT, j = 0,1,... (2.10) lim
C\: if Wj G Xj and w € X are solutions of the linear equations J*(*>,- = F?( Xj )v? F'( Zj )w = F"( Xj )v] where Xj, Zj G Dj, Vj € span(^ — Xj), then H-*iHI
Page 1 and 2: Konrad-Zuse-Zentrum für Informatio
Page 3 and 4: The authors wish to thank Erlinda C
Page 5 and 6: 1. A Refined Newton-Mysovskii Theor
Page 7 and 8: We note that under the hypothesis o
Page 9 and 10: Indeed (1.6) follows from (1.21) by
Page 11: 2. Asymptotic Mesh Independence of
Page 15 and 16: and by using property C 0 we obtain
Page 17 and 18: then for k > k(x°,e), where \x —
Page 19 and 20: Condition (2.30) is generally not s
Page 21 and 22: implies (2.12) — without any quas
Page 23: [12] L. B. Rail: A Note on the Conv

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