3D MHD Equilibria Using Local Spline Basis Functions

The problem of designing numerical solvers for finding 3D magnetohydrodynamic (MHD) equilibria has been heavily studied, and yet questions remain. Current research has focused on pseudo-spectral methods in order to solve the force balance equilibria equations, like those used in the code suite DESC. An important element in selecting pseudo-spectral methods over other techniques is their ease in providing the solution in a continuous and global manner. However, global spectral bases struggle to represent equilibria with non-smooth, sharp surfaces, which could pose a challenge for modelling quasi-isodynamic stellarators, divertors, and more realistic toroidal device geometry. To address these potential issues, we did the following: (1) implemented local spline basis functions on tetrahedral meshes, (2) successfully represented known 3D MHD equilibria in the local spline basis of tetrahedra (including local mesh refinement), and (3) used this new basis to perform stellarator shape optimization.