An Adaptive Newton-based Equilibrium Solver with Structure-Preserving Initialization for Dynamic MHD

We develop a Newton-based free-boundary Grad-Shafranov (GS) solver using adaptive finite elements and advanced preconditioning. The free-boundary interaction introduces a domain-dependent nonlinear form, with Jacobian contributions derived via shape calculus. Key innovations include the treatment of global constraints, nonlocal reformulations, and adaptive mesh refinement. The solver achieves robust convergence, reducing the nonlinear residual below 1e-6 within a few iterations, even in challenging cases where traditional Picard-based solvers fail.

To support dynamic MHD simulations, we analyze errors introduced when transferring GS equilibria to MHD discretizations. These errors, often stemming from mismatches in mesh alignment or function space choices, can degrade force balance and the divergence-free condition of the magnetic field. We identify critical factors affecting the quality of transferred equilibria, including mesh alignment and compatibility between GS and MHD function spaces. Numerical results show that structure-preserving choices substantially reduce initialization errors, maintain force balance, and weakly preserve magnetic divergence-free properties, enhancing the reliability of dynamic MHD instability studies in tokamaks.