Pousin, J.; Rappaz, J. Consistency, stability, a priori and a posteriori errors for Petrov- Galerkin methods applied to nonlinear problems. (English) Zbl 0822.65034 Numer. Math. 69, No. 2, 213-231 (1994). 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)]. Reviewer: W.C.Rheinboldt (Pittsburgh) Cited in 1 ReviewCited in 40 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators Keywords:nonlinear mappings; Petrov-Galerkin method; Banach spaces; consistency; stability; error estimates Citations:Zbl 0727.65087 PDF BibTeX XML Cite \textit{J. Pousin} and \textit{J. Rappaz}, Numer. Math. 69, No. 2, 213--231 (1994; Zbl 0822.65034) Full Text: DOI OpenURL