  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 an $H^{1/2}$-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 that each $H^{1/2}$-stable projection
yields convergence of the adaptive algorithm even with quasi-optimal
convergence rate. Numerical experiments with the $L^2$- and Scott-Zhang
projection conclude the work.
