A direct finite element method for elliptic interface problems

In this paper, a direct finite element method is proposed for solving interface problems on simple unfitted meshes. The fact that the two interface conditions form a $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair leads to a simple and direct weak formulation with an integral term for the mutual interaction over the interface, and the well-posedness of this weak formulation is proved. Based on this formulation, a direct finite element method is proposed to solve the problem on two adjacent subdomains separated by the interface by conforming finite element and conforming mixed finite element, respectively. The well-posedness and an optimal a priori analysis are proved for this direct finite element method under some reasonable assumptions. A simple lowest order direct finite element method by using the linear element method and the lowest order Raviart-Thomas element method is proposed and analyzed to admit the optimal a priori error estimate by verifying the aforementioned assumptions. Numerical tests are also conducted to verify the theoretical results and the effectiveness of the direct finite element method.