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

French, Donald A.; Larsson, Stig; Nochetto, Ricardo H.

Comput. Methods Appl. Math. 1, No. 1, 18-38 (2001).

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.

Reviewer: J.Lovíšek (Bratislava)

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)

Keywords:

finite element method; elliptic obstacle problem; a posteriori error estimate; maximum norm; penalty method; algorithm