Coupling-Aware Vanka Smoothing for Multigrid Preconditioning of the Implicit Immersed Boundary Equations

Abstract: The immersed boundary (IB) method models fluid--structure interaction using the natural Lagrangian and Eulerian formulations of structural mechanics and fluid dynamics, respectively, but explicit time discretization of the IB force imposes a stiffness-dependent upper bound on the time-step size. Treating these forces implicitly removes this restriction, but the resulting coupled linear systems become increasingly difficult to solve as the structural stiffness increases. Algebraically eliminating the Lagrangian degrees of freedom yields a reduced Eulerian velocity--pressure IB system. Here we introduce a coupling-aware Vanka (CAV) smoothing strategy to enable effective multigrid preconditioning of this system. CAV patches are built as unions of standard pressure-centered Vanka patches, with the graph of the Eulerian elasticity matrix determining which patches are combined. Under grid refinement, CAV patch sizes remain bounded, and the computational cost of each multigrid cycle thereby grows linearly with the number of Eulerian degrees of freedom. Tests using target-point, membrane, and beam force laws show that CAV-preconditioned FGMRES reduces the relative residual by ten orders of magnitude in $9--15$ iterations, with little growth under grid refinement. In a nonlinear benchmark modeling flow past a flexible fiber, the average number of FGMRES iterations per Newton solve increases only from $8.6$ to $9.5$ as the Eulerian grid is refined from $32\times32$ to $256\times256$ cells. To our knowledge, CAV provides the first robust multigrid strategy for time-dependent implicit IB formulations.