The adjoint analysis of plane Poiseuille flow is conducted for both global and convective stability problems. Receptivity and structural sensitivity are analyzed using a stabilized finite element method. In the global stability problem at high Reynolds numbers, the leading adjoint modes exhibit a boundary-layer-like structure at the inlet and near the channel walls. For convective stability problem, receptivity is highest in regions close to the walls similar to the global adjoint modes. Structural separation of direct and adjoint modes indicates high non-normality of the linear operators. Structural sensitivity analysis shows that both the global and convective stability are most sensitive to momentum perturbation near the wall and become increasingly sensitive to near-wall perturbations as Reynolds number increases. Slight variation of wall shape tends to affect the linear stability of plane Poiseuille flow significantly at high Reynolds numbers.