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## Section: New Results

### 2D Triangular Powell-Sabin Finite Elements

Participants : Giorgio Giorgiani, Hervé Guillard, Boniface Nkonga.

In order to avoid some mesh singularities when using quadrangular meshes for complex geometries and flux surfaces shapes, the use of triangular elements is a possible option that we are studying in view of its application to MHD modelling. It is not so easy to derive smooth finite element on triangle with reduced number of degree of freedom (ddl). The Bell reduced-quintic finite elements we have considered in the previous years have too much unknowns (6 per vertex). Powell-Sabin splines are piece-wise quadratic polynomials with a global C1-continuity and 3 unknowns per vertex, they have a local support, they form a convex partition of unity, they are stable, and they have a geometrically intuitive interpretation involving control triangles. In the previous years, we have developed the geometrical tools necessary to the construction of the Powell-Sabin splines and we are now beginning the study of the applicability of Powell-Sabin finite element for the numerical solution of PDE. We have used the Powell-Sabin starting from elliptic partial differential equations (including Grad-shafranov). We have applied these tools to solve hyperbolic 2D Euler equations with VMS stabilization. These results have been published in [11] and [18]