On the convergence of higher order finite element methods for nonlinear magnetostatics

TL;DR

This paper analyzes the convergence of higher-order finite element methods applied to nonlinear magnetostatics, providing theoretical proofs of existence, uniqueness, and optimal convergence rates, along with practical numerical results.

Contribution

It offers a comprehensive convergence analysis and proof of solution properties for higher-order finite element schemes in nonlinear magnetostatics, including nonlinear, anisotropic, and inhomogeneous materials.

Findings

Proven existence and uniqueness of solutions at continuous and discrete levels.

Established order optimal convergence rates under smoothness assumptions.

Demonstrated global linear and local quadratic convergence of the Newton solver.

Abstract

The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -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 further prove local quadratic convergence in a mesh-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.