We have developed a numerical algorithm based on 2D/3D finite element method for solving non‐Newtonian fluid flow with the moving free surface encountered in polymer processing. The power law model is considered as a rheological constitutive equation. The standard Galerkin finite element formulation/penalty formulation are applied to discrctize the governing equations, the volume‐of‐fluid (VOF) scheme is used to track the moving free surface, and the donor‐acceptor model introduced by Hirt and Nichols is modified and implemented on FEM. We applied the numerical scheme to simulate fountain flow and viscous buckling problems. For fountain flow, the numerical prediction of this study is in good agreement with the experimental results of other investigators. For viscous buckling, both 2D and 3D numerical simulations show that the shear thinning effect retards buckling. As this algorithm is very effective in treating moving free surface problems and requires less memory than previous algorithms, it may help solve challenging problems in polymer processing such as transient visroelastic flow simulations with moving free surfaces.