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M. Feischl, M. Page, D. Praetorius:
"Convergence of adaptive FEM for some elliptic obstacle problem with inhomogeneous Dirichlet data";
International Journal of Numerical Analysis and Modeling, 11 (2014), 1; 229 - 253.

In this work, we show the convergence of adaptive lowest-order FEM
(AFEM) for an elliptic obstacle problem with globally affine obstacle
and non-homogeneous Dirichlet data. 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 (2009) to the case of
non-homogeneous Dirichlet data and introduces some energy estimates to
overcome the lack of nestedness of the discrete spaces. A short
conclusion adresses AFEM for problems with non-affine obstacles.

http://www.math.ualberta.ca/ijnam/Volume11.htm