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Estimations a posteriori d’un schéma de volumes finis pour un problème non linéaire. (A posteriori estimates of a finite volume scheme for a nonlinear problem.). (English) Zbl 1033.65095
Consider problem (P):
Find \(u\in H^1(\Omega)\), solution of \(-\)div\((K(u)\nabla u)=f\) in the open bounded set \(\Omega\in \mathbb R^2\) for a given function \(f\in L^2(\Omega)\), with \(u=0\) on the Lipschitzian boundary \(\Gamma\), where \(K:\mathbb R\rightarrow \mathbb R\) is a nonlinear mapping such that, for all \(s\in \mathbb R\), \(0<\delta\leq K(s)\leq\alpha\) and \(| K'(s)| \leq\beta\), where \(\delta\), \(\alpha\) and \(\beta\) are constants.
Consider also hypothesis (H):
\(\sum_{\gamma\subset\partial V} | \gamma| \Phi(u_V^{ext},u_V^{int},n_\gamma)=\int_{\partial V}K(u_T)\nabla u_T\cdot n_V\,ds\) on the numerical flux function \(\Phi\).
The authors obtain a posteriori estimates of the error in approximating problem (P) by a finite volume method. These estimators are based on a reliable and efficient error indicator under hypothesis (H). This indicator is written in terms of the residual of the partial differential equation (PDE) in its strong form in every intersection between an element of the primal mesh and an element of the dual mesh and in terms of the jumps of the principal part of the PDE across the element boundaries.
This indicator can be applied to an arbitrary numerical flux function \(\phi\) if one can construct an interpolant \(u_T\) from the values \(u_V\) which verifies (H). This estimator can be used as a criterion for a mesh adaptation technique. The techniques developed in this work can be adapted to vertex-centered finite volume discretizations.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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