Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics (Advances in Applied Mathematics and Mechanics, 2018)
In this paper multigrid smoothers of Vanka-type are studied in thecontext of Computational Solid Mechanics (CSM). These smoothers wereoriginally developed to solve saddle-point systems arising in thefield of Computational Fluid Dynamics (CFD), particularly forincompressible flow problems. When treating (nearly) incompressiblesolids, similar equation systems arise so that it is reasonable toadopt the 'Vanka idea' for CSM. While there exist numerousstudies about Vanka smoothers in the CFD literature, only fewpublications describe applications to solid mechanical problems.With this paper we want to contribute to closing this gap. We depictand compare four different Vanka-like smoothers, two of them areoriented towards the stabilised equal-order $Q_1/Q_1$ finite elementpair. By means of different test configurations we assess how farthe smoothers are able to handle the numerical difficulties thatarise for nearly incompressible material and anisotropic meshes. Onthe one hand, we show that the efficiency of allVanka-smoothers heavily depends on the proper parameter choice. Onthe other hand, we demonstrate that only some of them areable to robustly deal with more critical situations. Furthermore, weillustrate how the enclosure of the multigrid scheme by an outerKrylov space method influences the overall solver performance, andwe extend all our examinations to the nonlinear finite deformationcase.