  One of the most well-established codes for modeling non-linear
Magnetohydrodynamics (MHD) for tokamak reactors is JOREK, which solves these
equations with a B\'ezier surface based finite element method. This code
produces a highly sparse but also very large linear system. The main solver
behind the code uses the Generalized Minimum Residual Method (GMRES) with a
physics-based preconditioner, but even with the preconditioner there are issues
with memory and computation costs and the solver does not always converge well.
This work contains the first thorough study of the mathematical properties of
the underlying linear system. It enables us to diagnose and pinpoint the cause
of hampered convergence. In particular, analyzing the spectral properties of
the matrix and the preconditioned system with numerical linear algebra
techniques, will open the door to research and investigate more performant
solver strategies, such as projection methods.
