The Finite Difference Method Of Removing Source Singularities For Three-Dimensional Elliptic Interface Problems

The immersed interface method is mainly used to solve partial differential equations with discontinuous interface, and now has been widely used in the field of computational fluid dynamics. The solution of partial differential equations derived interface problems are usually not continuous when they through the interface, which makes most routine numerical methods for partial differential equations are no longer applicable in solving immersed interface problems.The algorithm thinking of using the immersed interface method for elliptic interface problems is to use the undetermined coefficients method to determine the finite difference scheme at each grid point. There are some surface derivatives in the correction term derived from the difference scheme of an irregular point need to be computed. These quantities can be obtained by the least squares interpolation scheme. However, the implementation of these quantities will not only greatly increase the complexity of the algorithm, but is also not trivial for non-experts in this area. In order to reduce the amount of computation, we adopt a new strategy on the basis of the IIM. We extend the discontinuous jump conditions along the normal direction by level set function to construct a new function. Therefore, we can obtain a new differential equation that contains natural jump conditions, and the solvers of this new equation is continuous when they through the interface. We need not to compute any surface derivatives by using the discrete strategy for the new differential equation, and this is the main motivation for this extension and the transformation for the interface problems. Moreover, from this we can make the difference format simplified. Obviously, this approach can bring a greater advantage in solving the three-dimensional interface problems.In this paper we first describe the main steps of the IIM for elliptic interface problems, and then solve the three-dimensional immersed interface problems by removing source singularities. Finally, we use the fast Poisson solver to solve the finite difference equations and give practical examples to verify the reliability of the numerical results.