Multigrid on unstructured meshes with regions of low quality cells (2402.12947v1)

Abstract: The convergence of multigrid methods degrades significantly if a small number of low quality cells are present in a finite element mesh, and this can be a barrier to the efficient and robust application of multigrid on complicated geometric domains. The degraded performance is observed also if intermediate levels in a non-nested geometric multigrid problem have low quality cells, even when the fine grid is high quality. It is demonstrated for geometric multigrid methods that the poor convergence is due to the local failure of smoothers to eliminate parts of error around cells of low quality. To overcome this, a global--local combined smoother is developed to maintain effective relaxation in the presence of a small number of poor quality cells. The smoother involves the application of a standard smoother on the whole domain, followed by local corrections for small subdomains with low quality cells. Two- and three-dimensional numerical experiments demonstrate that the degraded convergence of multigrid for low quality meshes can be restored to the high quality mesh reference case using the proposed smoother. The effect is particularly pronounced for higher-order finite elements. The results provide a basis for developing efficient, non-nested geometric multigrid methods for complicated engineering geometries.