The finite element approximation of variational inequalities related to ergodic control problems (Q1361285)

The authors consider variational inequalities of the form \(a(u_{\alpha},v-u_{\alpha}) + \alpha\cdot (u_{\alpha},v-u_{\alpha})\geq (f,v-u_{\alpha})\), where \(u_{\alpha} \in H^{1}(\Omega)\), \(u_{\alpha} \leq\psi\); \(v \in H^{1}(\Omega)\), \(v\leq \psi\), \(\Omega\) is a given bounded smooth open set in \(\mathbb{R}^{N}\), \(f\) is a given function in \(L^{\infty}(\Omega)\), \(\psi\) is a positive obstacle of \(W^{2,\infty}(\Omega)\), \(a(u,v)=\int_{\Omega} \text{grad} (u)\cdot \text{grad} (v)dx\). Under suitable assumptions the solution \(u_{\alpha}\) of this inequality converges in the \(L^{\infty}\) norm to \(u_{0}\in H^{1}(\Omega)\), the unique solution of the inequality \((u_{0},v-u_{0})\geq (f,v-u_{0})\), as \(\alpha\) tends to zero, \(\alpha > 0\). Supposing that \(\Omega\) is a polyhedral, a regular quasi-uniform triangulation of \(\Omega\) into \(n-\)simplices of diameter less than \(h\) leads to the discrete variational inequality \(a(u_{\alpha h}v_{h}- u_{\alpha h})+\alpha \cdot (u_{\alpha h}v_{h} - u_{\alpha h}) \geq (fv_{h}-u_{\alpha h})\). Using some results of \textit{Ph. Cortey-Dumont} [Numer. Math. 47, 45-57 (1985; Zbl 0574.65064)], the authors prove the following error estimates: \(|u_{\alpha}-u_{\alpha h} |_{\infty} \leq C h^{2} |\log h|^{2}\), \(|u_{0} - u_{0h} |_{\infty} \leq Ch^{2} |\log h|^{2}\), where \(C\) is a constant independent of both \(\alpha\) and \(h\). NB. The paper is carelessly edited and typeset; in particular, the reference to the above mentioned paper of Cortey-Dumont, instead of to the 47th, points to the 5th volume of Numer. Math.