  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
