Year: 2014
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 1 : pp. 229–253
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 (2013) 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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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2014-IJNAM-523
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 1 : pp. 229–253
Published online: 2014-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Adaptive finite element methods Elliptic obstacle problems Convergence analysis.