## Section: New Results

### Stabilized finite element methods: interface problems

Participant : Roland Becker.

An accurate discretization method for incompressible elasticity or for the Stokes problem with varying, piecewise constant viscosity has been developed in [11] . This work is based on the NXFEM approach, initially developed in [53] and [54] for elliptic interface-problems and compressible elasticity, which gives a rigorous formulation of the very popular XFEM method known from crack-simulations in elasticity. In collaboration with Peter Hansbo, Chalmers Technical University (Sweden), and Erik Burman, University of Sussex (UK), we have been able to establish the inf-sup condition, necessary in the incompressible case, using stabilized P^{1}-P^{0} finite elements.
A typical computation with our method is shown in Figure 5 .

This research topic, which we wish to expand in different directions, such as robustness with respect to constants, adaptivity, and fast implementation, is related to the objectives of Concha within several respects. At a mature state, we expect that the proposed technology will be able to handle many problems with strongly heterogenous coefficients and data; it should also lead to a variational formulation of the so-called immersed-boundary method, allowing for rigorous error analysis and optimization algorithms.

One actual research direction is the development of shock-capturing methods for compressible flow problems based on NXFEM. The potential of this approach lies in the fact that local mesh-refinement could (at least partially) avoided, which is especially interesting for moving shocks. The free jet problem is an ideal test problem for this case.