This study presents an investigation of the plunge stage in joining AZ31B magnesium alloy with friction stir welding using two different 3D finite element models based on Arbitrary Lagrangian–Eulerian formulation and Coupled Eulerian–Lagrangian formulation. The investigations are made with the ABAQUS program. Johnson–Cook plastic material law and Coulomb friction law are used in both models. Models are compared in terms of temperature, strain distribution, and processing time. In both models, very similar temperature and strain distributions are obtained in the weld zone and the models are validated by experimental results. In addition, with the increase in the rotational speed of the tool, temperature and strain in the welding zone increase similarly in both models. In the model using the Arbitrary Lagrangian–Eulerian formulation, mesh distortions occur when high mesh density is not created in the plunge zone. No problems related to mesh distortion are encountered in the model using Coupled Eulerian–Lagrangian formulation. Moreover, it is found that the model using the Coupled Eulerian–Lagrangian formulation has a lower processing time and this processing time is not affected by the rotational speed of the tool.