Computable error estimates for linearization and numerical solution of obstacle problems (Q1891037)

The author studies the problem (1) \(\min\{E(v): v\in V\}\) and its linearized analog (2) \({\min\{E(v)}\); \(v\in H^ 1_{\Gamma_ 1, g_ 1}(\Omega)\}\), where \(E(v)= \int_ \Omega({1\over 2} | \nabla v|^ 2- fv) dx- \int_{\Gamma_ 2} g_ 2 vd\Gamma\), \(V= \{v\in H^ 1_{\Gamma_ 1, g_ 1} (\Omega)\mid v(x)\geq \psi(x)\), \(x\in \Omega\}\), \(H^ 1_{\Gamma_ 1, g_ 1} (\Omega)= \{v\in H^ 1(\Omega)\mid v(x)= g_ 1(x),\;x\in \Gamma_ 1\}\), \(\partial\Omega= \overline\Gamma_ 1\cup \overline\Gamma_ 2\), \(\Gamma_ 1\cap \Gamma_ 2= \emptyset\), \(\text{meas}(\Gamma_ 1)\neq 0\). Let \(u\) and \(u_ 0\) be solutions of (1) and (2), respectively. The main result of the paper is the following a posteriori error estimate for the linearization of the initial obstacle problem (1): \[ \|\nabla (\widetilde u_ 0- u)\|^ 2_{L^ 2(\Omega)}\leq \int_ \Omega (| \nabla \widetilde u_ 0|^ 2- | \nabla u_ 0|^ 2- 2f(\widetilde u_ 0- u_ 0)) dx- \int_{\Gamma_ 2} 2g_ 2(\widetilde u_ 0- u_ 0) d\Gamma, \] where \(\widetilde u_ 0= \max\{u_ 0, \psi\}\). A detailed analysis for a one-dimensional example is given.