Stochastic resistive Hall--MHD with current fluctuations: martingale weak solutions via a convergent structure-preserving finite element method

We study the stochastic resistive Hall–magnetohydrodynamic (Hall–MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall–MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.