This review paper discusses the developments in immersed or unfitted finite element methods over the past decade. The main focus is the analysis and the treatment of the adverse effects of small cut elements. We distinguish between adverse effects regarding the stability and adverse effects regarding the conditioning of the system, and we present an overview of the developed remedies. In particular, we provide a detailed explanation of Schwarz preconditioning, element aggregation, and the ghost penalty formulation. Furthermore, we outline the methodologies developed for quadrature and weak enforcement of Dirichlet conditions, and we discuss open questions and future research directions.

翻译：本审查文件讨论了过去十年来沉浸或不适合的有限要素方法的发展动态,主要重点是分析和处理小切割要素的不利影响,我们区分对系统条件稳定的不利影响和对系统条件的不利影响,我们概述已开发的补救办法,特别是我们详细解释施瓦兹的先决条件、要素汇总和幽灵刑罚的提法,此外,我们概述了为迪里切特条件的二次调整和执行不力而制定的方法,我们讨论了开放的问题和今后的研究方向。