Preconditioning for finite element methods with strain smoothing

TL;DR

This paper develops and analyzes preconditioning techniques for strain smoothing finite element methods, improving their efficiency for large-scale problems by leveraging spectral properties and proposing a two-level additive Schwarz preconditioner.

Contribution

It introduces a novel preconditioning approach based on spectral analysis, including an improved two-level additive Schwarz preconditioner for strain smoothing methods.

Findings

Spectral analysis of stiffness matrices reveals key properties.

The proposed preconditioner accelerates iterative solutions.

Numerical experiments confirm theoretical improvements.

Abstract

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.