Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.

翻译：过去二十年间，解为$C^0$连续的椭圆界面问题已得到充分研究。经典数值方法包括强稳定广义有限元法（SGFEM）和浸入式有限元法（IFEM）。本文数值研究了一类更广泛的椭圆界面问题，其解具有不连续性。直接应用现有方法会导致失败，因为近似解所处的函数空间更大，需覆盖不连续函数。我们提出了一类高阶增强非拟合有限元法，用于求解具有隐式或Robin型界面跳跃条件的问题。我们设计了新型增强函数，既能捕捉解中强制的不连续性，又能控制条件数快速增长。近期已有一维线性增强方法利用单一增强函数实现，我们将其推广至任意高阶，采用两个简单的单侧不连续增强函数。通过自然张量积扩展至二维情形的可行性得到验证。在$L^2$范数和分片$H^1$范数下建立了最优阶收敛性，并证明了所有离散节点处的超收敛性（特殊情形下可精确还原节点值）。数值算例验证了理论分析。最后，为证明该方法在实际问题中的有效性，将线性、二次和三次增强单元应用于药物洗脱支架的多层壁模型，该模型同时包含零通量跳跃条件和隐式浓度界面条件。