  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.
