A Mixed Finite Element Simulation For Second Order Elliptic Interface Problems

The interface problems are often governed by a second order elliptic problem with discontinuous coefficients in material sciences and fluid dynamics when two or more distinct materials or fluids with different conductivities or densities of diffusions are involved. These interface problems must satisfy interface jump conditions due to conservation laws. When the interface is smooth enough, the solution of the interface problems are very smooth in individual regions occupied by materials of fluids, but due to the interface or the jump conditions, the global regularity of its solution is usually very low and only has order of Hl+a,0≤a< 1. This results in difficulties to achieve high accuracy for the standard numerical methods.For the following elliptic interface problem together with the jump conditions on the interface (?) we propose a mixed finite element method based on the interface-fitted grids in this paper. In this paper the normal continuity of the vector-functions in mixed finite element space will characterize exactly the flux jump conditions when crossing the edge between two adjacent elements. The numerical analysis shows that the mixed finite element method approximates the solution and the flux optimally and has the order of O(hk+1), which is important in the practical engineering. But in the practical engineering applications, the flux vectorβu is much more often used than the unknown function u is. The numerical analysis and numerical experiments conducted in this paper shows that the method approximates the solution and the flux optimally and is a robust method to model the interface problem.In view of the limits of the interface-fitted grids, we consider the effectiveness of the mixed finite element method with the unfitted grids. From the numerical ex-periments, we can find that the. mixed finite element method to solve.the interface problem is effective. Comparing to the above method, the order is lower.