2D Fourier finite element formulation for magnetostatics in curvilinear coordinates with a symmetry direction

We present a numerical method for the solution of three-dimensional linear magnetostatic problems by embedding their geometry into symmetric domain and Fourier expansion in symmetry coordinate, leading to set decoupled two-dimensional problems. Special cases include axial translational symmetry. To keep treatment universal, framework decomposition vector potential formulation is developed covariant formalism with general curvilinear coordinates. Under restriction homogeneous material parameters along direction, but not on fields current densities, this leads equations both, zeroth (non-oscillatory) non-zero (oscillatory) harmonics. For latter it possible eliminate one component resulting fully transverse orthogonal magnetic field. In addition Poisson-like equation longitudinal non-oscillatory problem, curl-curl Helmholtz describes problem covering oscillatory case. The variational forms are treated usual nodal finite element edge problem. can be computed independently each harmonic using fewer degrees freedom than simulation at comparable accuracy. These properties make approach useful analysis perturbation harmonics plasma confinement devices cylindrical coordinates or flux validated analytical test applied racetrack-shaped coil setup.