The focus of this study is on the treatment of those stellar regions immediately adjacent to convective zones . The results of hydrodynamical simulations by Freytag et al . ( 1996 ) show that the motion of convective elements extends well beyond the boundary of the convectively unstable region . We have applied their parametrized description of the corresponding velocities to the treatment of overshoot in stellar evolution calculations up to the AGB ( Pop . I , M _ { ZAMS } = 3 M _ { \sun } ) . Our calculations show the 3 ^ { rd } dredge-up already at the 7 ^ { th } thermal pulse ( TP ) , and the dredge-up parameter reaches \lambda = 0.6 during the next five pulses . Accordingly , the amount of dredged up ^ { 12 } C is up to 10 ^ { -3 } M _ { \sun } . Our models develope a small so-called ^ { 13 } C-pocket consisting of a few 10 ^ { -7 } M _ { \sun } . Finally , this treatment of boundaries of convective regions leads to intershell abundances of typically ( ^ { 4 } He/ ^ { 12 } C/ ^ { 16 } O ) = ( 23/50/25 ) ( compared to ( 70/26/1 ) in the standard treatment ) .