Fast Solvers for 3D Poisson Equations Involving Interfaces in an Finite or the Infinite Domain

摘要

In this paper, numerical methods are proposed for Poisson equations defined in a finite or in the infinite domain in three dimensions. In the domain, there can exists an interface across which the flux and the solution are discontinuous. To deal with the discontinuity in the source terms and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can he carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the niesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can he solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms.