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The presentation will summarize the recent developments and open problems in the numerical techniques for free-boundary and fluid-structure interaction problems. The techniques under consideration are based on a finite element spatial discretization and time-splitting. The fluid-structure interaction methods are based on the so-called fictitious domain approach (see Glowinski et 2001). There are several particular issues that make the application of splitting methods to free-boundary problems quite awkward. First, the very essence of the splitting methods is in the separation of the pressure and velocity which are coupled through the so-called generalized Stokes problem. In case of Dirichlet boundary conditions this splitting can be shown to have an optimal splitting error for the velocity and slightly suboptimal error for the pressure (in case of a second order scheme). However, in case of open boundary/free boundary conditions, the splitting can yield a large suboptimal splitting error for both, velocity and pressure, due to their coupling on the free boundary via the force ballance condition (see Guermond et al., 2004). Improving this splitting is still an open question and we will show some ideas for further work. Probably the most widely used approach for solving free-boundary and fluidstructure interaction problems is the so-called Eulerian approach. It raises the other major concern that will be discussed in this presentation. While moving the boundaries through the stationary grid, it generally intersects some elements. Most of the currently available techniques ignore this and use the usual (locally very smooth) interpolation. This is in contrast with the actual properties of the solution across the interface and significantly worsens the interpolation error. Some possible remedies of this problem within the framework of the finite element method will also be demonstrated. The third issue that we would like to consider is about the solution of the pure advection problem for the motion of the free-boundary. There are several ways for tracking the free boundary including the level-set, the volume-of-fluid and the surface tracking algorithms. But a major problem for each of them when combined with splitting methods is that the end-of-step velocity of these methods is not divergence free and this makes the advection problem non-conservative which in turn leads to a significant mass loss. A projection method that tries to circumvent this issue will be presented. In addition, some numerical results illustrating the methods under consideration will be also presented. REFERENCES
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