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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)
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