Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations
For researchers using immersed boundary methods, this work provides scalable solvers that improve parallelization opportunities over prior Vanka-type smoothers, though performance degrades at high Reynolds numbers and for thin interfaces.
This work develops parallelizable smoothing strategies (additive Vanka and Schur complement-based smoothers) for a geometric multigrid preconditioner for the immersed boundary equations, achieving effective linear solver performance for fluid-structure interaction problems, particularly at lower Reynolds numbers and modest-to-high elastic stiffnesses.
The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically, we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.