## Pointwise A Posteriori Error Analysis For An Adaptive Penalty Finite Element Method For The Obstacle Problem

By Donald A. French, Stig Larsson, Ricardo and Ricardo H. Nochetto

### Abstract

. 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. 1. Introduction We consider finite element approximations of the obstacle problem \Gamma\Deltau(x) + fi(u(x) \Gamma /(x)) 3 f(x); x 2\Omega ; u(x) = 0; x 2 @\Omega ; (1.1) where / and f are given functions with / 0 on @ and where fi is the maximal monotone graph defined by fi(s) = 8 ? ! ? : f0g; s ? 0; (\Gamma1; 0]; s = 0; ;; s ! 0: (1.2) Our analysis and our finite element method are based on the following penalized (or regularized) form of (1.1): find u ffl such that \Gamma\Deltau ffl (x) + ffl(x) \Gamma1 (u ffl (x) \Gamma /(x)) \Gamma = f(x); x 2\Omega ; u ffl (x) = 0; x 2 @\Omega ; (1.3) w..

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