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 - <p style="text-align: justify;">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 &amp; 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 &amp; Hoppe (2007)
and Page &amp; 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.</p>