Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem
Abstract
This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the -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 -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.
Cite
@article{arxiv.1811.09260,
title = {Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem},
author = {Dominik Hafemeyer and Christian Kahle and Johannes Pfefferer},
journal= {arXiv preprint arXiv:1811.09260},
year = {2018}
}