The regularization method for an obstacle problem (Q1347055)

The problem under consideration is to find \(u_ 0\in H^ 1_ 0(\Omega)\): \[ a(u_ 0, v- u_ 0)+ j(v)- j(u_ 0)\geq \ell(v- u_ 0)\quad (v\in H^ 1_ 0(\Omega)),\tag{1} \] where \(a(u,v)= \int_ \Omega \nabla u \nabla v dx\), \(j(u)= \int_ \Omega \varphi(u)dx\), \(\ell(u)= - \int_ \Omega \nabla g\nabla u dx\), \(\varphi(t)= | t+ g|\). The nondifferentiable term \(j(u)\) in (1) is approximated by the smooth functional \(j_ \varepsilon(u)= \int_ \Omega \varphi_ \varepsilon(u)dx\) with \(j_ \varepsilon(u)\to j(u)\) \((u\in H^ 1_ 0(\Omega))\) as \(\varepsilon\to 0\), \(\varepsilon> 0\). In the usual way it is shown that \(u_ \varepsilon\to u_ 0\) strongly as \(\varepsilon\to 0\), where \(u_ \varepsilon\) denotes the solution of the regularized analog of (1) with \(j_ \varepsilon\) instead of \(j\). With the using of the duality technique the following a posteriori error estimate is established: \[ \textstyle{{1\over 2}} \| \nabla(u_ \varepsilon- u_ 0)\|^ 2_{L^ 2(\Omega)}\leq \int_ \Omega (| u_ \varepsilon|- u_ \varepsilon (\varphi_ \varepsilon)' (u_ \varepsilon))dx. \] Similar inequalities are also proved for the case when the above mentioned regularization is combined with discretization of (1) by the finite element method.