The authors treat the solution of three-dimensional Poisson equations defined in a finite \((\Omega^{-})\) or an infinite domain \(\Omega^{+}\) with a finite jump in the flux of the solution across the interface of \(\Omega ^{-}\) and \(\Omega^ {+}\). The simulation of ferromagnetic material is mentioned as an application. The interface problem with a discontinuous/nonsmooth solution is transformed to a problem with a smooth solution.
The existence and uniqueness of the solution is proved. To solve the problem numerically the infinite domain is separated using an auxiliary sphere centered at the origin with the radius \(r = a\) and applying Kelvin’s inversion to transform the Poisson equation in the exterior unbounded domain to an equation in the bounded interior domain. The two equations are coupled at the boundary \(r = a\). Choosing the mesh carefully a centered second-order finite difference approach for spherical coordinates is used to discretize the equations. Applying the fast Fourier transform (FFT) the difference equations are transformed into a block tridiagonal system of linear algebraic equations which is solved using a cyclic reduction method. The Poisson equation is also solved in a sphere \(r\leq a\) using artificial boundary conditions with half of the work computing only the solution inside the truncating boundary. The methods are validated by examples.