Pointwise a posteriori error analysis for an adaptive penalty finite element method for the obstacle problem (Q2723224)

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.NEWLINENEWLINENEWLINEMain result: The error bound in the maximum norm of the form NEWLINE\[NEWLINE\|U_\varepsilon- u_\varepsilon\|_{L_\infty(\Omega)}\leq C|\log h_{\max}|\|h^2 R_\infty\|_{L_\infty(\Omega)}NEWLINE\]NEWLINE and the penalty error in the case of a smooth obstacle function NEWLINE\[NEWLINE\Psi\in W^2_\infty(\Omega):\|u- u_\varepsilon\|_{L_\infty(\Omega)}\leq \|\varepsilon(f+ \Delta\Psi)\|_{L_\infty(\Omega)},NEWLINE\]NEWLINE are proposed.