Stability and convergence of the penalty formulation for nonlinear magnetostatics
<|reference_start|>Stability and convergence of the penalty formulation for nonlinear magnetostatics: The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem<|reference_end|>
@article{egger2024stability,
    title={Stability and convergence of the penalty formulation for nonlinear
  magnetostatics},
    author={Herbert Egger and Felix Engertsberger and Klaus Roppert},
    journal={arXiv preprint arXiv:2403.18285},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.18285},
    primaryClass={math.NA cs.NA math.OC}
}