Magnetic relaxation drives plasma toward lower-energy equilibria under helicity constraints. In ideal MHD, helicity is locally conserved; in resistive settings like Taylor relaxation, only global helicity is preserved, permitting local reconnection. We present two structure-preserving finite element methods based on finite element exterior calculus and Lagrange multipliers, which enforce helicity conservation at these two different levels. Applied to the magneto-frictional system (gradient flow of MHD), our schemes preserve a discrete Arnold inequality and maintain nontrivial magnetic topology over a long time. Numerical experiments on braided and knotted fields confirm that local helicity preservation prevents spurious reconnection, while global-only conservation allows further relaxation. We then apply the relaxation idea to the MHD turbulence models to preserve ideal invariants and capture the selective decay mechanism, which is believed to be a mechanism leading to the formation of large magnetic eddies.