Because of its great adaptability, the Arbitrary Lagrangian Eulerian (ALE) method is often used to solve the Navier-Stokes equations with a free surface. The kinematic condition relating the normal velocity to the mesh velocity suggests a simple way to move the domain, but it leads to unstable schemes in many cases. A method to take into account the non-linearity of the free surface is presented in this paper and integrated into a Finite Method with Galerkin characteristics. A variational form of the surface tension is used to overcome the difficulty of estimating the main curvature of the surface grid. A stability estimate is then established on the global scheme, under regularity conditions on the grid.
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Bertrand Maury (1996) studied this question.
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