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arXiv · 1811.09260

Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem

Dominik Hafemeyer·Christian Kahle·Johannes Pfefferer
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Abstract

This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of point-wise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global $L^2$-error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.

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BibTeXRIS

Dominik Hafemeyer, Christian Kahle, Johannes Pfefferer. 2018-11-22. Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem. https://arxiv.org/abs/1811.09260

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