## 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)].

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators

Zbl 0727.65087
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