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## Section: New Results

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

Participants : Francesca Rapetti, Holger Heumann.

Existing finite element implementations for the computation of free-boundary axisymmetric plasma equilibria approximate the unknown poloidal flux function by standard lowest order continuous finite elements with discontinuous gradients. The location of critical points of the poloidal flux, that are of paramount importance in tokamak engineering, is constrained to nodes of the mesh, which leads to undesired jumps in transient problems. Moreover, recent numerical results for the self-consistent coupling of equilibrium with resistive diffusion and transport suggest the necessity of higher regularity when approximating the flux map.

In [23], we have proposed a mortar element method that employs two overlapping meshes. One mesh with Cartesian quadrilaterals covers the vacuum domain and one mesh with triangles discretizes the region outside the vacuum domain. The two meshes 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 a mortar-like projection. We have shown that the numerical calculation of free boundary plasma equilibria highly benefits from approximating the poloidal flux through some higher regular FE functions in the interior of the limiter.

In [19], we have rather analysed the precision of the proposed approach, by varying the discretization parameters. We thus compute the approximation error between the computed and the synthetic solution of a model problem for the same method adopted in [1], by changing, for example, the local polynomial degree in the subdomains, the size of the overlap between the meshes, the local size of the mesh elements. Indeed, FE methods on composite meshes are widely used in practice, but their theoretical foundation is fairly limited in the literature. Therefore, we have reported in [2] some experimental convergence results for different discretization schemes involving composite meshes.