We study a class of enriched unfitted finite element or generalized finite element methods (GFEM) to solve a larger class of interface problems, that is, 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. The major challenge of GFEM development is to construct enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. The linear stable generalized finite element method (SGFEM) was recently developed using one enrichment function. We generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. So is the optimal order convergence at all nodes. To prove the efficiency of the SGFEM, 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：我们研究的是一组浓缩的不合格有限要素或通用有限要素方法(GFEM),以解决更大规模的接口问题,即1D异端界面问题与不连续的解决方案,包括具有隐含或罗宾式界面跳跃条件的问题;GFEM发展的主要挑战是建立浓缩功能,既能捕捉解决方案的不连续性,又能保持快速增长的条件数;线性稳定的一般有限要素方法(SGFEM)最近使用一种浓缩功能开发;我们利用两个简单的单向式浓缩功能将其推广到任意程度;建立了以2美元为单位的优化组合,并打破了1美元-诺姆;所有节点的最佳顺序趋同也是最佳的组合;为了证明SGFEM的效率,将浓缩的线性、四方和立方元素应用到一个多层墙模型,用于毒品食用区,其中均存在零氟跳动条件和隐含浓度界面条件。