Section: New Results

Finite element methods for interface problems

Participants : Nelly Barrau, Roland Becker, Robert Luce.

The original formulation of NXFEM [63] is based on the doubling of elements. In some situations, as the case of a moving interface, it is computationally more convenient to have a method with local enrichment, as for the standard XFEM. In [47] we have developed such an approach based on NXFEM. We have developed an hierarchical formulation for a fictitious domain formulation in [7] .

One of the technical difficulties is the simultaneous robustness of the method with respect to the size of the intersection of a mesh cell with the interface and with respect to the discontinuous diffusion parameters. In [ ] (note CRAS 2012) we proposed a modified formulation of the NXFEM which allows us to obtain this robustness to solve the Darcy equation.

In connection with the thesis of Nelly Barrau, supervised by Robert Luce and Eric Dubach (LMAP) we have:

• implemented lots of geometrical tools in 2D and 3D necessary to use the NXFEM methods,

• extended the method to $P_k$ and $Q_k$ finite elements ([42] ,

• generalized the residual estimator and developed an adaptative process with hanging node (8 ),

• adapted the method to the transport equation.

Figure 8. Result of an adaptative process with hanging node