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Pointwise error estimates for a class of elliptic quasi-variational inequalities with nonlinear source terms. (English) Zbl 1065.65082
The aim of the paper is to show that the class of elliptic quasi-variational inequalities (QVIs) with nonlinear source terms can be properly approximated by a finite element method, provided some realistic assumptions are made on both the nonlinearity \(f(\cdot)\) and the operator \(S\).
Main results: The author show that the class of QVIs \((a(u, v- u)\geq(f (u), v- u)\) for any \(v\in{\mathbf K}(u)= \{v\in H^1_0(\Omega): v\leq S(u)\) a.e. in \(\Omega\}\), \(S(u)= \psi+\phi(u)\), \(\psi\) is a regular function, \(\phi(u)\) is a nonlinear operator from \(L_\infty(\Omega)\) into itself such that: \(\phi(w)\leq\phi(\widetilde w)\) for \(w\leq\widetilde w\) a.e. in \(\Omega\), \(\phi(w+ C)\leq\phi(w)+ C\) \((C> 0)\), \(f(\cdot)\) is a Lipschitz nondecreasing nonlinear source term with rate \(\alpha\), \((\alpha/\beta) <1)\) can be properly approximated by a finite element method, when the approach rests on a discrete \(L_\infty\)-stability result and \(L^\infty\)-error estimates for elliptic QVIs.
The continuous QVI problem is associated with a fixed point mapping and the existence of a unique continuous solution is proved. A discrete stability result for elliptic linear QVIs is established and an abstract \(L_\infty\)-error estimate is proposed. The extension to the noncoercive case is considered, too. Finally, the applications to an obstacle type problem and to an impulse control problem are given.

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49M15 Newton-type methods
Full Text: DOI
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