## Section: New Results

### Finite Element Methods

#### Gyrokinetic quasi-neutrality equation

Participants : Nicolas Crouseilles, Eric Sonnendrücker.

In [21] , a new discretization scheme of the gyrokinetic quasi-neutrality equation is proposed. We discretised the gyrokinetic Poisson equation using arbitrary order spline finite elements which enables to accommodate more complex domains. Moreover in standard polar coordinates we developed a fast solver which is comparable in computational time to the original FFT-second order finite differences, but can become more efficient for higher order as fewer grid points are needed for the same accuracy.

#### Dissipative boundary conditions for finite element codes

Participants : Philippe Helluy, Laurent Navoret, Eric Sonnendrücker.

We are developing finite-element codes for the Vlasov-Poisson system that would be able to capture the filamentation phenomenon. The filamentation is a mechanism that transfers the space fluctuations of the distribution function to high frequency oscillations in the velocity direction. For stability purpose, most numerical schemes contain dissipation that may affect the precision of the finest oscillations that could be resolved. In [60] , [61] , [62] Eliasson constructs a non reflecting and dissipative condition for the Fourier-transformed Vlasov-Poisson system. The condition enables the high velocity-frequency oscillations to leave the computational domain in a clean way.

We are currently developing a finite-element code based on this dissipative boundary condition. The code is part of the Selalib library. We also propose an approximation of the Eliasson method, based on the Béranger's PML formalism. Contrary to the original boudary conditions that requires a space Fourier transformation, this method is local and thus could be extended to higher dimensionnal problems and more complex geometries.

#### High order finite element methods for Maxwell

Participants : Stéphanie Salmon, Eric Sonnendrücker.

In paper [23] , we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes. We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes. We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.