## Section: New Results

### Preconditioning for elliptic solvers

Participants : Emmanuel Franck, Mariarosa Mazza, Ahmed Ratnani, Eric Sonnendrücker, Stefano Serra-Capizzano.

The different algorithms to discretize in time the MHD or to design preconditioning use solvers for a lot of elliptic operators like Laplacian. For high order finite elements like B-Splines the classical multi-grid methods are not very efficient. Indeed the number of iterations to converge increases strongly when the polynomial order increases. Using a theory called GLT, proposed by S. Serra-Capizzano, we have implemented and validated a smoother for multi-grid, able to obtain the convergence quasi independent of the polynomial degree. This method is also efficient as a preconditioning for mass matrices. We obtain at the end, very robust solvers for these simple problems and allows to perform the time algorithm for fluid models. The next step is to extend this method for more complex problems like vectorial elliptic problems.