Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element~(SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods. First, we analyze the spectrums of the stiffness matrices of the edge-based S-FEM and the SSE method. Then, we propose an improved two-level additive Schwarz preconditioner for the strain smoothing methods by modifying local solvers appropriately. For the sake of convenience of implementation, an alternative form of the preconditioner is also proposed by defining the coarse-scale operation in terms of the standard FEM. We verify our theoretical results through numerical experiments.

翻译：平滑的方法,如平滑的有限元素方法(S-FEM)和抽压的元素方法(SSE)成功地改善了有限元素的性能,并且在有限元素分析中应用了这些元素。为了对大规模问题有效应用,必须开发出一个数学和数字完善的超高频解答器,用于缓解松动方法。在本文中,根据松动方法的光谱特性,我们提出了高效的预设方法。首先,我们分析了以边缘为基的S-FEM和SESE方法的僵硬性矩阵的频谱。然后,我们提出了改进的双级添加剂Schwarz 先决条件,通过适当修改当地溶剂来缓解松动方法。为了便于实施,还提出了一种先决条件的替代形式,即根据标准的FEM定义粗略的操作。我们通过数字实验来核实我们的理论结果。