This talk will include joint work with a number of colleagues in the Schools of Computing, Mathematics and Mechanical Engineering at the University of Leeds: P.H. Gaskell, M.A. Kelmanson, R.C. Peterson, J.L. Summers, H.M. Thompson, M.A. Walkley and M.C.T. Wilson.

We describe a very general ALE algorithm, using a boundary- conforming finiteelement method, for the solution of surface-tension-dominated free-surface flow problems in two and three dimensions. This algorithm combines both mesh movement, driven by the motion of the free surface, with discrete remeshing, if and when the quality of the mesh deteriorates below certain quantified thresholds. The main strength of the adaptive approach that has been adopted is that it allows the shape of the free-surface to change significantly during the course of a simulation without any a priori knowledge of how the geometry will evolve. Furthermore, the use of adaptivity ensures that the freesurface may be represented to a high accuracy and the ALE approach means that the computational domain is restricted to the region occupied by the fluid at any given time. Our implementation using triangles in two dimensions, allowing both Cartesian and axisymmetric flow simulations, is well developed, whilst the three-dimensional implementation based upon tetrahedra is still ongoing. Details of the adaptive algorithms and the underlying flow solvers will be presented. In particular we will describe how the position of the free-surface is updated at each time step and then how the positions of the rest of the nodes in the mesh are moved, and the finite element equations updated, to be consistent with this. The mechanisms for triggering discrete remeshing will also be discussed, as will the remeshing algorithms themselves and the data structures required for efficient interpolation between grids after discrete remeshing has taken place. Example simulations that will be described include the shedding of a curtain of fluid from a rotating roller, the formation of an axisymmetric drop, the breaking of a liquid bridge as two wet surfaces are pulled apart, and the viscous sintering of two cylinders or two spherical droplets of fluid. In each of these cases the geometry of the solution domain changes substantially during the course of the simulation and so the use of mesh adaptivity is essential for an efficient computation to be achieved.