Multigrid on unstructured meshes with regions of low quality cells

TL;DR

This work analyzes the degradation of geometric multigrid convergence on non-nested, unstructured meshes caused by small regions of low-quality cells. It introduces a global--local smoother that couples a standard global smoother with local, direct-solvers applied on small, low-quality subdomains, forming a symmetric Schwarz-type preconditioner. Through Poisson and linear elasticity experiments in 2D and 3D, the method restores the high-quality convergence rate, often outperforming standard smoothers, particularly for higher-order elements. The approach enhances robustness and suggests viable pathways for efficient, non-nested geometric multigrid solvers on complex engineering geometries, with potential extensions to AMG.

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.

Figures (19)

• Figure 1: Unstructured grid of the unit square with a poor quality region near the centre.
• Figure 2: Relative residual at the end of each multigrid cycle for high and low quality meshes. Symmetric Gauss--Seidel is used as the smoother.
• Figure 3: A (a) high frequency Fourier mode as initial guess, and (b) absolute value of the error at each vertex after five iterations of symmetric Gauss--Seidel.
• Figure 4: Absolute value of error on each vertex after five iterations of Jacobi-preconditioned Chebyshev method on the low quality unit square model problem.
• Figure 5: Domain decomposition view of the combined global--local smoother, which consists of the global smoother $S_g$ on the whole domain and the local residual correction $S_c$ on the locally poor quality subdomains.