Gravity-driven continuous thin liquid film flow over a flexible membrane containing surface topography is modelled using lubrication theory. The associated coupled nonlinear equation set, for the film thickness, pressure and membrane deflection, is solved using a state-of-the-art full approximation storage (FAS) multigrid algorithm with automatic mesh refinement and adaptive time-stepping, in order to maximise computational efficiency when fine-scale resolution is required while ensuring accurate mesh independent solutions at the micro-scale. The robustness of the approach is demonstrated through the solution of a series of problems and comparisons drawn with the same flow on an equivalent completely rigid membrane. It is shown that the former differs considerably from the latter in that the film thickness affects the shape of the flexible membrane, the compliance of which in turn impinges on the profile of the resulting free-surface disturbance
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