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\(L^\infty\)-error estimate for a discrete two-sided obstacle problem and multilevel projective algorithm. (English) Zbl 1115.65070

The authors investigate an \(L_\infty (\Omega)\)-error estimate for the finite element approximation of a two-sided obstacle problem: Find \(u \in K(\Omega)\) such that \(a(u,v-u)\geq (f,v-u)_{L_2 (\Omega)}, \forall v \in K(\Omega)\) where \(K(\Omega)\), a closed convex set in \(H^1_0(\Omega)\) has the form: \(K(\Omega)=\{v \in H^1_0(\Omega):\Phi\leq v\leq \Psi\) in \(\Omega \}\), \(\Omega \subset \mathbb{R}^2\) is a bounded convex polygon, \(\Phi, \Psi \in W^{2,s}(\Omega)(s>2)\) are two given functions such that \(\Phi | _{\bar{\Omega}}<\Psi | _{\bar{\Omega}}\) and \(\Phi | _{\partial \Omega}<0<\Psi | _{\partial \Omega}\), \(a(u,v)=\int_\Omega \nabla u \cdot \nabla v \;dx, \;f \in L^S (\Omega)\), by using the results of variational inequalities with one obstacle.
Main result: The rate of convergence is \(O(h^2|\log h|)\), provided \(\Phi , \Psi \in W^{2,\infty}(\Omega)\) and \(f \in L_\infty (\Omega)\) is established. The authors show that the order of convergence is the same as that of variational inequalities with one obstacle. Finally a multilevel projective algorithm is proposed and its convergence is discussed.

MSC:

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