Aggregation-Based Algebraic Multigrid Fixed-Point Solvers for Nonlinear Magnetostatics in Linear Time: Scalability Testing on TEAM Problem 13

The robustness and computational efficiency of the φ- and A-methods for nonlinear magnetostatic problems are investigated using the TEAM Problem 13 benchmark. The φ-method, formulated in terms of nodal variables, employs fixed-point iterations accelerated by the AGMG algebraic multigrid solver for div-grad systems. The A-method, formulated in terms of edge variables, combines the secant method with AGMG CC, a specialized AGMG variant designed for curl-curl linear systems. Numerical results show that both methods exhibit robust convergence and linear-time solution complexity for magnetic media with monotonic apparent permeability, but stagnate or fail in the non-monotonic case. Since replacing multigrid with a preconditioned conjugate gradient solver does not improve convergence, the observed behavior can be attributed to the fixed-point iteration itself. This finding reveals a previously unreported limitation of fixed-point approaches for nonlinear magnetostatic problems and suggests that alternative nonlinear solvers may be required for materials with non-monotonic apparent permeability.