## Section: New Results

### A finite element method with overlapping meshes for free-boundary toroidal plasma equilibria in realistic geometry

Participants : Holger Heumann, Francesca Rappetti.

Existing finite element implementations for the computation of free-boundary toroidal plasma equilibria approximate the flux function by piecewise polynomial, globally continuous functions. Recent numerical results for the self-consistent coupling of equilibrium and resistive diffusion in the spirit of Grad-Hogan suggest the necessity of higher regularity. Enforcing continuity of the gradient in finite elements methods on triangular meshes, leads to a drastic increase in the number of unknowns, since the degree of the polynomial approximation needs to be increased beyond four. Therefore existing implementations for the fixed boundary problem resort to (curvilinear) quadrilateral meshes and approximation spaces based on cubic Hermite splines. Fine substructures in the realistic geometry of a tokamak, such as air-gaps, passive structures and the vacuum vessel prevent the use of quadrilateral meshes for the whole computational domain, as it would be necessary for the free-boundary problem.

In this work we propose a finite element method that employs
two meshes, one of quadrilaterals in the vacuum domain and one of triangles outside,
which *overlap* in a narrow region around the vacuum domain.
This approach gives the flexibility to achieve easily and at
low cost higher order regularity for the approximation of the flux function in the domain covered by
the plasma, while preserving accurate meshing of
the geometric details exterior to the vacuum.
The continuity of the numerical solution in the region of overlap is weakly enforced by relying on the mortar projection.
A publication is in preparation.