  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.
