Volume 11, Issue 1
Convergence of Adaptive FEM for Some Elliptic Obstacle Problem with Inhomogeneous Dirichlet Data

Int. J. Numer. Anal. Mod., 11 (2014), pp. 229-253.

Published online: 2014-11

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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 (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.

• Keywords

Adaptive finite element methods, Elliptic obstacle problems, Convergence analysis.

65N30, 65N50

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@Article{IJNAM-11-229, author = {}, title = {Convergence of Adaptive FEM for Some Elliptic Obstacle Problem with Inhomogeneous Dirichlet Data}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {1}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/523.html} }
TY - JOUR T1 - Convergence of Adaptive FEM for Some Elliptic Obstacle Problem with Inhomogeneous Dirichlet Data JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 229 EP - 253 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/523.html KW - Adaptive finite element methods, Elliptic obstacle problems, Convergence analysis. AB -

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.

M. Feischl, M. Page & D. Praetorius. (1970). Convergence of Adaptive FEM for Some Elliptic Obstacle Problem with Inhomogeneous Dirichlet Data. International Journal of Numerical Analysis and Modeling. 11 (1). 229-253. doi:
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