## Section: New Results

### ${C}^{1}$ finite elements on triangular meshes

Participants : HervĂ© Guillard, Ali Elarif, Boniface Nkonga.

In order to avoid some mesh singularities that arise when using quadrangular elements for complex geometries and flux aligned meshes, the use of triangular elements is a possible option that we have studied in the past years. However due to the appearence of fourth order terms in the PDE systems that we are interested in, pure Galerkin methods require the use of finite element methods with ${C}^{1}$ continuity. The PhD thesis of Ali Elarif that has begun in october 2017 is devoted to the study of these methods for complex PDE models encountered in plasma physics. Relying on the work previously done on steady elliptic PDE, this year we applied these finite element methods to some evolution problems like the incompressible Navier-Stokes and MHD equations in stream-function formulation. Error estimates in ${H}^{2}$ norms have been obtained using standard finite element techniques. The simulation of some instabilities encountered in plasma physics have been done with very satisfactory results.