The Finite Difference Method Of Three-dimensional Elliptic Interface Problems

The finite difference method is mainly used for the solution of partial differential equations with discontinuous interface problems. And now has been widely used in the field of computational fluid dynamics. Second order accurate elliptic solvers using Cartesian grids are presented for three dimensional interface problems in which the source term, the coefficients, the solution and its normal flux may be discontinuous across an interface. This would make the most of numerical methods can not be better applied in these models. In the paper, the applications of immersed interface method in three dimensional elliptic is studied, the main contributions are as following.1. For the general three-dimensional elliptic immersed interface problems, the algoritgm thinking of using the immersed interface method for such models is to using the level set function represents the interface, the mesh points are divided into irregular points and regular points, and then establish a new coordinate system. In the new coordinate system,using the undetermined coefficients method to determine the finite difference scheme at each grid point, then derived the correction in the irregular points and discuss the different situation. Thus, we get a global second-order finite difference.2. For three dimensional elliptic immersed interface problems, when its coefficient is piecewise constant, according to the basic idea of this kind of model is based on the fast immersed interface method and a fast Poisson solution. Solving process, we use GMRES iteration methods. And presents some practical example. By combining the numerical experiment results show the feasibility of the method.