## Section: New Results

### Dirichlet-to-Neumann vs Bayliss-Gunzburger-Turkel operators for the modelling of elliptical scatterers at low and mid frequency

Participants : Hélène Barucq, Rabia Djellouli, Anne Gaëlle Saint-Guirons.

The solution of a scattering problem generally involves the coupling of the physical model with Absorbing Boundary Conditions (ABC). The efficiency of such conditions has a big impact on the accuracy of the numerical solution. Thus their construction is one of the main step of the numerical method handling. If the ABC design is generally based on high frequency assumptions, numerical investigations illustrate the fact that most of the ABC are valid at low frequency also. In [30] , the effect of the wave number on the performance of local ABC has been done for the cases of the circle and the sphere. In this work we construct a new family of Dirichlet-to-Neumann (DtN) conditions for elliptical shaped scatterers which is quite general for applications in scalar acoustic problems. The conditions apply to a scalar acoustic problem governed by the Helmholtz equation. The DtN maps are designed to be local because we aim at keeping the sparsity of the discrete finite element matrix. Then we analyze their performance at low and mid frequencies by considering in a first time the radiator problem both for the two and the three-dimensional cases. This analysis allows us to select the best DtN condition and is based on a modal analysis involving Mathieu functions (2D) [21] or spheroidal wave functions (3D) [29] . In a second time, we do the same comparisons by considering the scattering problem which is solved by using the On-Surface-Radiation-Condition method [36] . We can also enhance the effect of the eccentricity on the performance of the conditions. Next, we complete the theoretical study by comparing the efficiency of the DtN operators with Bayliss-Gunzburger-Turkel (BGT) operators which were formerly studied by Reiner et al. [42] . The comparison is done for the radiator and the scattering problems. Our conclusions are the following. For a thin ellipse (the eccentricity is close to 1), the second-order DtN condition outperforms the second-order BGT one while both conditions tend to give rise to similar results for the sphere (the eccentricity is zero). The supremacy of the DtN condition is more and more obvious as the frequency increases to begin really significant at high frequency.