We use two fourth order geometric partial differential equations to efficiently solve several surface modelling problems, including the surface blending, the N-sided hole filling and the free-form surface fitting with the G1 boundary continuity. The nonlinear equations used include the surface diffusion flow and the Willmore flow. These nonlinear equations are discretized using the mixed finite element method based on the combination of the Loop?s basis and the linear basis. The proposed approach is simple, efficient and gives very desirable results.