A direct finite element method for elliptic interface problems

In this paper, a direct finite element method is proposed for solving interface problems on unfitted meshes. This new method treats the two interface conditions as an $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair for the mutual interaction across the interface, rather than the jumps of variables. A simple and straightforward finite element method is proposed based on this approach. This method solves the interface problem using conforming finite elements in one subdomain and conforming mixed finite elements in the other, with a natural integral term accounting for mutual interaction. Under reasonable assumptions, this direct finite element method is proved to be well-posed with an optimal a priori error analysis. Moreover, a simple lowest-order direct finite element method, using the linear element and the lowest-order Raviart-Thomas element, is analyzed to achieve the optimal a priori error estimate by verifying the aforementioned assumptions. Numerical tests are provided to confirm the theoretical results and the effectiveness of the direct finite element method.