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

M. Feischl, M. Page & D. Praetorius

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.

  • AMS Subject Headings

65N30, 65N50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-229, author = {M. Feischl, M. Page and D. Praetorius}, 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 AU - M. Feischl, M. Page & D. Praetorius 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 and D. Praetorius. (2014). 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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