FINITE ELEMENT CENTER PREPRINT 2000–12 APosteriori Error Analysis in themaximumnorm for a penalty finite element method for the time- dependent obstacle problem

A Posteriori Error Analysis in the maximum norm for a penalty finite element method for the time-dependent obstacle problem Abstract. We consider nite element approximation of the parabolic obstacle problem. The analysis is based on a penalty formulation of the problem where the penalisation parameter is allowed to vary in space and time. We estimate the penalisation error in terms of the penalty parameter and the data of the equation. The penalised problem is discretised in space and time by means of a Discontinuous Galerkin method. We prove an a posteriori error estimate in the space-time maximum norm involving a residual and the stability property of a linearised adjoint problem. 1. introduction In this note we study numerical solution of the time dependent obstacle problem by means of a nite element method. Our work is based on a penalty formulation of the problem and concerns an a posteriori error estimate in the maximum norm. In our context the penalty method consists of the introduction of a penalised problem, a certain nonlinear partial diierential equation involving a penalty parameter , whose solution converges to the solution of the time dependent obstacle problem as tends to zero. The penalty problem is approximated by means of a nite element method. Using this approach Scholz 13], 14] proved optimal a priori error estimates in the energy norm, for the stationary and the time dependent obstacle problems. Optimal error estimates in the L 2 norm are not known. Recently French, Larsson and Nochetto 5] proved an a posteriori error estimate in the maximum norm for the stationary obstacle problem. In the present work we prove an a posteriori error estimate in the maximum norm for the time dependent obstacle problem. Our analysis allows the penalty parameter to vary in space and time, which might be useful in adaptive algorithms. In our method there are two sources for the error. The rst part, the penalisation error, comes from the use of the penalty problem. This part is estimated in maximum norm, in terms of the penalty parameter and data, using an a priori estimate. The second part, the discretisation error, comes from the nite element discretisation of the penalty problem. We use a Discontinuous Galerkin method, see 2] and 3], to discretise the penalty problem