A simple and efficient method for solving open boundary axisymmetric magnetic field problems is proposed. This technique uses two ellipsoidal analysis domains which are connected through the periodic boundary conditions. One of the domains acts as an exterior region which produces no reflection of low-order multipoles. The harmonic solutions of the Laplace equation in an oblate spheroidal coordinate have been investigated and a formula for the axis ratio of the ellipsoidal boundary and the permeability of the exterior region has been derived. To verify the formula, we have demonstrated some numerical examples and compared with spectrum-domain solutions. Based on the results, we concluded that the methodology is proven to be valid.