The reduced scalar potential in regions with permeable materials: Reasons for loss of accuracy and cancellation

Practical three-dimensional magnetic field problems usually involve regions containing current sources as well as regions with magnetic materials. For computational purposes the use of the reduced scalar potential (RSP) as unknown has the advantage to transform a problem for a vector field throughout the space into a problem for a scalar function, thus reducing the number of degrees of freedom in the discretization. However in regions with high magnetic permeability the use of the RSP alone usually results in severe loss in accuracy and it is recommended to use both the reduced scalar potential and the total scalar potential. Using an asymptotic expansion, we investigate theoretically the underlying reasons for this lack of accuracy in permeable regions when using the RSP as a unique potential. Moreover this investigation leads to an efficient numerical method to compute the magnetic field in regions with high magnetic permeability.