Simulation of thin film flows with a moving mesh mixed finite element method

We present an efficient mixed finite element method to solve the fourth-order thin film flow equations using moving mesh refinement. The moving mesh strategy is based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To achieve a high quality mesh, we adopt an adaptive monitor function and smooth it based on a diffusive mechanism. A variety of numerical tests are performed to demonstrate the accuracy and efficiency of the method. The moving mesh refinement accurately resolves the overshoot and downshoot structures and reduces the computational cost in comparison to numerical simulations using a fixed mesh.