The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit residual-based error estimator as well as a separate marking strategy based on the alternative error estimator and an optimal data approximation algorithm. This paper reviews and discusses available convergence results. In addition, all three strategies are investigated empirically for a set of benchmarks examples of second-order elliptic partial differential equations in two spatial dimensions. Particular interest is on the choice of the marking and refinement parameters and the approximation of the given data. The numerical experiments are reproducible using the author's software package octAFEM available on the platform Code Ocean.

翻译：最小二乘有限元方法的收敛性分析催生了多种自适应网格细化策略：基于内置后验误差估计器或替代显式残差型误差估计器的集体标记算法，以及基于替代误差估计器与最优数据逼近算法的分离式标记策略。本文综述并讨论了现有收敛性结果。此外，针对二维空间中二阶椭圆偏微分方程的一组基准算例，对所有三种策略进行了实证研究。特别关注标记与细化参数的选择以及给定数据的逼近问题。数值实验可通过作者在 Code Ocean 平台上提供的软件包 octAFEM 进行复现。