On the Convergence of Higher-Order Finite Element Methods for Nonlinear Magnetostatics

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 55-75, February 2026.
Abstract. The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree–dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.