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Reliable solution for a 1D quasilinear elliptic equation with uncertain coefficients. (English) Zbl 0944.35027
The paper deals with the existence of reliable solutions (the notion introduced by Hlavàček) for the quasilinear elliptic equation \[ -(a(u) u')' = f \quad \text{in } \Omega , \qquad u = \overline{u} \quad \text{on } \Gamma_1, \qquad a(u) u' = g \quad \text{on } \Gamma_2 \] where \(\Omega = (0,1), \Gamma_1 = \{0\} \text{ (or } \{0,1\}), \Gamma_2 = \{0,1\} \setminus \Gamma_1, a\) is a Lipschitz continuous function on \(\mathbb R\), \(\overline u\) is a constant. The reliable solution we are looking for (\(u \in C^1(\overline{\Omega}) \cap C^2(\Omega)\)) depends on the coefficient function which is not known exactly; it is uncertain and belongs to an admissible set. The reliable solution is ”the worst” case among the set of possible solutions, the “badness” is measured by a cost functional. The Kirchhoff transformation is applied to obtain the existence of the state solutions and a cost sensitivity formula. The problem is approximated by means of a finite element method. Some convergence results are proven and numerical examples are given.

35J65 Nonlinear boundary value problems for linear elliptic equations
35R60 PDEs with randomness, stochastic partial differential equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
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