Stability of algebraic multigrid for Stokes problems

SUMMARY

An investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle-point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed-order and equal-order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh-dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed-order interpolation to permit stable smoothing and therefore to deliver mesh-independent convergence is explained. Two expedient options are suggested for obtaining mesh-independent convergence for those AMG codes that are based on an equal-order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.

The stability of smoothing in algebraic multigrid solvers, when applied to strongly coupled field problems (saddle-point problems), is shown to depend on the degree of coarsening in the coarse-grid approximation (CGA). Stability is most fragile for CGAs constructed with inter-grid transfer operators based on an equal-order interpolation of the scalar and vector fields. The ability of CGAs based on mixed-order interpolation to provide for a stable smoothing, and hence a mesh-independent convergence, is explained.