The paper under review deals with a stabilized finite element discretization in relationship with the strong convergence of discrete magnetization fields with reduced convergence order for a uni-axial model problem. In the first part of the paper, the Euler-Lagrange equations for the relaxed minimization problem are deduced and a priori estimates are established. Next, the authors prove the strong \(L^2\)-convergence for discrete solutions of the stabilized problem. Numerical experiments are presented in the final section.