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arXiv · 1207.3257

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

Michael Feischl·Marcus Page·Dirk Praetorius
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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.

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Michael Feischl, Marcus Page, Dirk Praetorius. 2012-07-13. Convergence of adaptive FEM for some elliptic obstacle problem with inhomogeneous Dirichlet data. https://arxiv.org/abs/1207.3257

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