The convergence of standard multigrid methods decays significantly if locally poor quality cells are present, and it is found that the poor convergence is due to the local failure of the smoothing property. The high frequency error localised in regions of low quality cells is not eliminated by standard multigrid smoothers and persists through multigrid cycles. We propose a global–local combined smoother for the geometric multigrid to deal with engineering meshes with a small number of poor quality cells, which includes two steps: a global smoother on the whole domain, followed by a local correction on the subdomains with low quality cells. The high frequency error remaining in the low quality regions can be damped out completely by the local correction.
The idea is extended to the algebraic multigrid (AMG), including both classical AMG and smoothed aggregation AMG. It is suggested that the high frequency error produced by the smoother propagates outward the low quality region on the fine grid to the neighbouring areas on the coarse grid. An algorithm to track low quality regions on the abstract coarse grid of AMG has been developed based on the information transfer between grid levels via the transfer operators. With the local correction applied on low quality regions tracked on the abstract coarse grid, the high frequency error due to low grid quality can be removed. In the smoothed aggregation AMG, the construction of the smoothed prolongation operator depends on the spectral radius of the system. However, regions of low quality cells in a mesh increase the largest eigenvalue of the linear system. We propose a shifted largest eigenvalue strategy to approximate a reasonable spectral radius to construct the smoothed prolongation.
Two and three dimensional numerical experiments, from illustrate to complicated, are demonstrated to validate the proposed smoother. Elliptic type PDEs, including Poisson and elasticity problems, are solved. For each example, the performance of multigrid on a high quality mesh is also presented as a reference case, and it is shown that the poor convergence of multigrid for low quality meshes can be recovered to the reference case by the proposed smoother. A realistic thermomechanical simulation of turbomachinery problem has also been successfully solved
