A fast algorithm for particle simulations
Journal of Computational Physics
SIAM Journal on Numerical Analysis
The immersed interface method: a numerical approach for partial differential equations with interfaces
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
SIAM Journal on Scientific Computing
A Fast Iterative Algorithm for Elliptic Interface Problems
SIAM Journal on Numerical Analysis
Numerical solution of problems on unbounded domains. a review
Applied Numerical Mathematics - Special issue on absorbing boundary conditions
SIAM Review
Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients
SIAM Journal on Scientific Computing
Accurate numerical methods for micromagnetics simulations with general geometries
Journal of Computational Physics
New Formulations for Interface Problems in Polar Coordinates
SIAM Journal on Scientific Computing
Spectral method for matching exterior and interior elliptic problems
Journal of Computational Physics
A coupling interface method for elliptic interface problems
Journal of Computational Physics
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In this paper, numerical methods are proposed for Poisson equations defined in a finite or infinite domain in three dimensions. In the domain, there can exists an interface across which the source term, the flux, and therefore the solution may be discontinuous. The existence and uniqueness of the solution are also discussed. To deal with the discontinuity in the source term and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can be 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 mesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can be 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.