This study examines an effect of pressure up to 50 GPa on the elastic and mechanical properties of wurtzite gallium nitride (w-GaN) by using classical potential within the Atomistic Tool Kit (ATK)-force field. The obtained results show that the elastic constants and other related parameters, such as Young’s modulus, shear modulus, bulk modulus, Poisson’s ratio, Pugh’s ratio, Zener anisotropy factor and Kleinman parameter increase monotonically with increase of pressure up to 32 GPa. Beyond this pressure, we observed a non-linear behavior with increase in pressure. This might be attributed to the phase transition in GaN in the pressure range of 33.4 - 44.6 GPa. The results obtained for zero pressure are consistent with both experimental data and the theoretical data shown in references.