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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.

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