Motivated by the probe experiment for characterizing the adhesion strength of polymeric materials, we studied the axisymmetric extensional flow of a viscoelastic liquid filament that contains one or three initially spherical bubbles along its axis of symmetry. The filament is confined between two parallel and coaxial disks of the same radius and its initially cylindrical outer surface is surrounded by air. The flow is induced by the axial extension of the upper disk under constant velocity, while the lower disk remains stationary. The rheology of the liquid is described by the exponential Phan-Thien and Tanner viscoelastic model. A quasi-elliptic transformation combined with domain decomposition and local mesh refinement is employed to discretize the domain. The mixed finite element Galerkin method for the mass and momentum balances combined with the EVSS-G method and SUPG weighting for the constitutive equation are used to obtain their weak forms. All these are necessary for the successful simulation of much larger filament extensions and of liquids with higher elasticity than those reported earlier. The evolution of the filament and the bubble(s)-free surfaces depends on the interplay of the viscous, elastic, and capillary forces. It was found that, when the ratio of the elastic and the capillary forces is small compared to the viscous forces, the bubble attains very large deformations. The onset of cusps was observed at the bubble poles for intermediate values of capillarity and elasticity. The force varies with the filament extension (or time) in a way reported in experiments measuring the strength of adhesive materials: Its initial increase up to a maximum is followed by a plateau, and finally it drops to zero when the adhesive tends to fail. Increasing the filament elasticity delays the development of stresses (and the applied force).