Convergence of adaptive FEM for some elliptic obstacle problem with inhomogeneous Dirichlet data

TL;DR

This paper proves the convergence of an adaptive finite element method for elliptic obstacle problems with non-homogeneous Dirichlet data, extending previous analyses to more general boundary conditions and obstacles.

Contribution

It extends convergence analysis of adaptive FEM to obstacle problems with non-homogeneous Dirichlet data and non-affine obstacles, introducing new energy estimates.

Findings

Convergence of adaptive FEM for obstacle problems with inhomogeneous Dirichlet data.

Extension of residual-based error estimator to control Dirichlet data oscillations.

Establishment of a contraction property involving energy error, estimator, and oscillations.

Abstract

In this work, we show the convergence of adaptive lowest-order FEM (AFEM) for an elliptic obstacle problem with non-homogeneous Dirichlet data, where the obstacle \chi\ is restricted only by \chi\ in H^2(\Omega). The adaptive loop is steered by some residual based error estimator introduced in Braess, Carstensen & Hoppe (2007) that is extended to control oscillations of the Dirichlet data, as well. In the spirit of Cascon et al. (2008), we show that a weighted sum of energy error, estimator, and Dirichlet oscillations satisfies a contraction property up to certain vanishing energy contributions. This result extends the analysis of Braess, Carstensen & Hoppe (2007) and Page & Praetorius (2010) to the case of non-homogeneous Dirichlet data as well as certain non-affine obstacles and introduces some energy estimates to overcome the lack of nestedness of the discrete spaces.