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.

翻译：本文针对简单非贴合网格上的界面问题，提出了一种直接有限元法。两个界面条件构成$H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$对的事实，导出了一个简单直接的弱形式，其中包含界面上相互作用的积分项，并证明了该弱形式的适定性。基于此形式，提出了通过一致有限元和一致混合有限元在由界面分隔的两个相邻子域上求解问题的直接有限元法。在合理假设下，证明了该直接有限元法的适定性和最优先验分析。通过使用线性单元法和最低阶Raviart-Thomas单元法，提出并分析了一种简单的最低阶直接有限元法，通过验证上述假设获得了最优先验误差估计。数值实验也验证了理论结果和直接有限元法的有效性。