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**Consistency, stability, a priori and a posteriori errors for Petrov- Galerkin methods applied to nonlinear problems.**
*(English)*
Zbl 0822.65034

Let \(X\) and \(Y\) be reflexive Banach spaces and \(F: X\mapsto Y'\) a \(C^ 1\)-mapping from \(X\) into the dual \(Y'\) of \(Y\). For the problem (1) \(\langle F(u)| v\rangle= 0\), \(\forall v\in Y\), where \(\langle\cdot |\cdot \rangle\) is the duality pairing, consider approximate problems of the form (2) \(\langle F(u_ h)| v_ h\rangle= 0\), \(\forall v_ h\in Y_ h\), where \(X_ h\subset X\), \(Y_ h\subset Y\) are given subspaces such that \(\dim X_ h= \dim Y_ h< \infty\). A solution \(u\) of (1) is assumed to exists for which the FrĂ©chet derivative \(DF(u)\) is an isomorphism from \(X\) to \(Y'\).

Under consistency and stability conditions which, in essence, are linked to approximation properties of \(X\) by \(X_ h\) and \(Y\) by \(Y_ h\), and to discrete “inf-sup” conditions on the bilinear form \(\langle DF(u)\phi| \psi\rangle\), it is proved that (2) has a unique solution \(u_ h\) in a neighbourhood of \(u\). Moreover, a priori and a posteriori error estimates are established in the norm of \(X\). The principal term in the a posteriori estimate is the residual \(\| F(u_ h)\|_{Y'}\) in the \(Y'\)-norm. The main results have been announced by the authors [C. R. Acad. Sci., Paris, Ser. I 312, No. 9, 699-703 (1991; Zbl 0727.65087)].

Under consistency and stability conditions which, in essence, are linked to approximation properties of \(X\) by \(X_ h\) and \(Y\) by \(Y_ h\), and to discrete “inf-sup” conditions on the bilinear form \(\langle DF(u)\phi| \psi\rangle\), it is proved that (2) has a unique solution \(u_ h\) in a neighbourhood of \(u\). Moreover, a priori and a posteriori error estimates are established in the norm of \(X\). The principal term in the a posteriori estimate is the residual \(\| F(u_ h)\|_{Y'}\) in the \(Y'\)-norm. The main results have been announced by the authors [C. R. Acad. Sci., Paris, Ser. I 312, No. 9, 699-703 (1991; Zbl 0727.65087)].

Reviewer: W.C.Rheinboldt (Pittsburgh)

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47J25 | Iterative procedures involving nonlinear operators |