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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 Fréchet 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 |