The collapse of a spherical cavitation bubble contained in a large body of upper convected Maxwell fluid is theoretically analyzed by using a variational principle approach in the Lagrangian frame for the K-BKZ rheological equation of state with potential functions. Based on the Rayleigh time scale for bubble collapse in ideal fluids, two parameters, Re and De, are identified. Using a finite element technique, a fully explicit numerical scheme is developed both for the pressure distribution calculation and for bubble surface tracking. The same problem is formulated also using the Galerkin-finite element method in the Lagrangian frame for the differential model of an upper convected Maxwell fluid. With the latter method, the viscoelastic stress can be determined explicitly. Even though the result is the same as far as the radius-time curve is concerned, each method has its own advantages. Highly oscillatory behaviors in bubble radius are observed for moderate Re and De. For large ReDe, the solution exhibits an asymptotic behavior. It is also observed that fluid elasticity accelerates the collapse in the early stage of collapse while in the later stages it retards the collapse. The retardation for a moderate range of Re is expected to be related to the reduced cavitation damage in viscoelastic fluids.