The Immersed Interface Method For Solving The Interface Problem Of Elliptic Equations

Interface problems have many applications in life, for examples, the modeling of Stefan problem of solidification process, crystal growth, and composite materials. In the process of solving the interface problems, there must be some discontinuous or singularities present in the coefficients of the differential equations which are established reference to the physical background and the constraint conditions. The common methods are used to solve the problems whose solutions are smoothly unavailable to the interface problems. In view of this, many researchers put forward some methods dealing with interface problems. The immersed interface method is a finite difference method which combined the jump conditions of interface to correct the difference scheme and based on the Cartesian grid to ensure the accuracy of the finite difference discrete. The immersed interface method proposed by Li Zhilin leads to second-order global accuracy for the computed solution. What’s more, many difficult interface problems can be treated by this method.In this paper, an overview and comments on the immersed interface method were presented. First present methods solving the interface problems were expounded, then the basic principle of the immersed interface method was introduced briefly, and some application examples were listed and solved by the immersed interface method showing this method is validity and accuracy, finally the future investigation on the immersed interface method were recommended.