## Pointwise a posteriori error analysis for an adaptive penalty finite element method for the obstacle problem.(English)Zbl 0980.65118

Finite element approximations based on a penalty formulation of the elliptic obstacle problem are analyzed in the maximum norm. A posteriori error estimates, which involve a residual of the approximation and a spatially variable penalty parameter, are derived in the cases of both smooth and rough obstacles. An adaptive algorithm is suggested and implemented in one dimension.
Main result: The error bound in the maximum norm of the form $\|U_\varepsilon- u_\varepsilon\|_{L_\infty(\Omega)}\leq C|\log h_{\max}|\|h^2 R_\infty\|_{L_\infty(\Omega)}$ and the penalty error in the case of a smooth obstacle function $\Psi\in W^2_\infty(\Omega):\|u- u_\varepsilon\|_{L_\infty(\Omega)}\leq \|\varepsilon(f+ \Delta\Psi)\|_{L_\infty(\Omega)},$ are proposed.

### MSC:

 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
Full Text: