A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential for a narrow band level set method, is introduced based on a finite element $L^2$- or $H^1$-projection combined with the ghost-penalty method. This procedure is formulated as a linear variational problem in a narrow band around the surface, making it computationally efficient and suitable for rigorous error analysis. The extension method is combined with a discontinuous Galerkin space discretization and a BDF time-stepping scheme. The paper analyzes the stability and accuracy of the extension procedure and evaluates the performance of the resulting narrow band finite element method for the level set equation through numerical experiments.

翻译：本文提出了一种有限元方法，用于追踪由水平集方程控制的界面演化。该方法在界面周围的窄带区域内求解水平集指示函数。针对窄带水平集方法所必需的延拓过程，本文提出了一种基于有限元$L^2$-或$H^1$-投影并结合虚罚方法的实现方案。该延拓过程被表述为界面周围窄带内的线性变分问题，使其计算高效且适用于严格的误差分析。该延拓方法与间断伽辽金空间离散化及BDF时间步进格式相结合。本文分析了延拓过程的稳定性与精度，并通过数值实验评估了所得水平集方程窄带有限元方法的性能。