Global and local helicity-preservation in the finite element discretization of magnetic relaxation

Magnetic relaxation drives plasma toward lower-energy equilibria under helicity constraints.

In ideal magnetohydrodynamics (MHD), helicity is locally conserved, while resistive theories such as Taylor relaxation preserve only global helicity.

This distinction has important implications for structure-preserving numerical methods.

We compare three finite element formulations: an unconstrained scheme that does not conserve helicity, a mixed method based on finite element exterior calculus that preserves discrete local helicity on magnetically closed subdomains, and a Lagrange multiplier approach that enforces only global helicity conservation.

Numerical experiments with magnetic knots and braids show that helicity-based constraints provide effective topological barriers when the relevant helicity-type invariant is nonzero, but do not fully characterize braided field-line topology when it vanishes.

These results clarify both the strengths and the possible limitations of helicity-based structure-preserving finite element methods for magnetic relaxation.