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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.

MSC:
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
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