Abstract.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 ∈ 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 ≥ 1. 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 Scott-Zhang projection conclude the work.Mathematics Subject Classification. 65N30, 65N50.