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M. Aurada, M. Feischl, J. Kemetmüller, M. Page, D. Praetorius:
"Each H1/2-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd";
M2AN Math. Model. Numer. Anal., 47 (2013), 04; S. 1207 - 1235.

We consider the solution of second order elliptic PDEs in $\R^d$ with inhomogeneous Dirichlet data by means of an $h$-adaptive FEM with fixed polynomial order $p\in\N$. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of a stable projection, for instance, the $L^2$-projection for $p=1$ or the Scott-Zhang projection for general $p\ge1$. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments conclude the work.

http://dx.doi.org/10.1051/m2an/2013069