We present implementation details and analyze convergence of a two-grid solver forming the core of the fully automatic hp-adaptive strategy for elliptic problems. The solver delivers a solution for a fine grid obtained from an arbitrary coarse hp-grid by a global hp-refinement. The classical V-cycle algorithm combines an overlapping block Jacobi smoother with optimal relaxation, and a direct solve on the coarse grid. A simple theoretical analysis is illustrated with extensive numerical experimentation.