Z-mag: A Two-Chart Stereographic Formulation for Finite-Element Micromagnetic Simulations

Accurate enforcement of the unit-length constraint on magnetization is a central challenge in computational micromagnetics. Existing coordinate choices involve trade-offs: Cartesian formulations require explicit normalization or constraint stabilization; spherical coordinates have pole singularities at physically reachable states; and single-chart stereographic coordinates avoid finite-point singularities but become poorly scaled near the projection pole. In this work, we present Z-mag, an open-source finite-element solver based on a unified two-chart stereographic atlas that preserves the unit constraint by construction while maintaining robust conditioning across the full set of magnetization states. The method uses two stereographic charts with antipodal poles and a single sign-parameterized set of equations, avoiding separate chart-specific formulations. Around the equatorial band where both charts are valid and well-scaled, the solver blends their contributions with continuous atlas weights, avoiding hard chart switching during time evolution. Numerically, Zmag couples the Landau–Lifshitz–Gilbert magnetization dynamics with open-boundary magnetostatics through an operator-split scheme that combines Newton–Krylov nonlinear solves with a finite-element/boundary-element magnetostatic solve. Adaptive timestepping with thermodynamic acceptance criteria controls nonphysical transients and improves robustness in stiff dynamical regimes. Implemented with FEniCSx/DOLFINx and PETSc, Z-mag is designed for reproducible, parallel micromagnetic simulation.