Jiang, Ying-Jun; Zeng, Jin-Ping \(L^\infty\)-error estimate for a discrete two-sided obstacle problem and multilevel projective algorithm. (English) Zbl 1115.65070 Lobachevskii J. Math. 24, 43-53 (2006). 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. Reviewer: Jan Lovíšek (Bratislava) MSC: 65K10 Numerical optimization and variational techniques 49J40 Variational inequalities 49M15 Newton-type methods Keywords:variational inequality; two-sided obstacle problem; \(L^\infty\)-error estimate; order of convergence; multilevel projective algorithm; finite element PDFBibTeX XMLCite \textit{Y.-J. Jiang} and \textit{J.-P. Zeng}, Lobachevskii J. Math. 24, 43--53 (2006; Zbl 1115.65070) Full Text: EuDML EMIS